The trunk open indicator is lit, the trunk is closed and the chime keeps sounding even with the keys removed

Sounds like you have a bad switch in the latch. You can try spraying it with contact cleaner and operating the switch several times. But most likely you will need to change out the bad latch.

What is it chiming ? Chimes for different reasons , headlamps on , seat belt not buckled , etc.... What warning light is lit on the instrument cluster ? if any . seat belt , air bag's etc...

Audible Warnings Description and Operation
The audible warnings alert the driver of a system concern or a critical vehicle condition. The following lists the audible warning priority and the pulse rate:
Fast rate chime (200 pulses per minute)
Medium rate chime (150 pulses per minute)
Slow rate chime (50 pulses per minute)
Single chime
Fasten Safety Belt Warning
The inflatable restraint sensing and diagnostic module (SDM) activates the fasten safety belt audible warning for 5-8 seconds. The fasten safety belt warning sounds and the fasten seat belt indicator illuminates when the following occurs:
• The ignition switch is in RUN, BULB TEST, or START position.
• The SDM detects that the driver's seat belt is not buckled (signal is low).
If the seat belt is buckled before the ignition is in RUN, BULB TEST, or START position, the chime does not sound. If the seat belt is buckled while the chime is sounding, the chime stops.
Key-In-Ignition Warning
The instrument panel cluster (IPC) activates the key-in-ignition audible warning. The key-in-ignition warning sounds when the following occurs:
• The ignition switch is in OFF, LOCK, or ACCY position.
• The body control module (BCM) determines that the driver's door is open (signal is low).
• The BCM determines that the key-in-ignition switch is closed (signal is low).
Lights On Warning
The instrument panel cluster (IPC) activates the lights on audible warning. The lights on warning sounds when the following occurs:
• The key is not in the ignition.
• The body control module (BCM) determines that the drivers door is open (signal is low).
• The BCM determines that the headlamp switch is in the park or head position.
Rear Parking Assist
The rear object sensor control module activates the audible warning. The audible warning sounds whenever any of the indicators are illuminated on the rear parking assist display. For a description of the rear parking assist system refer to Object Detection Description and Operation.
Turn Signal Reminder
The instrument panel cluster (IPC) sounds the turn signal reminder once per each flash of the turn signal lamps when:
• The ignition switch is in the RUN position.
• The turn signal switch is in either turn position, sensing, within 1 second, an open to B+ transition in the turn signal control circuit (left or right turn).
• The vehicle has traveled a distance of 1.2 km (0.75 mi) by counting pulses on the vehicle speed input circuit.
The IPC turns off the turn signal reminder when either the ignition switch is turned to the LOCK position or to the OFF position, or the turn signal switch returns to the OFF position, sensing, within 1 second, no transitions from open to B+ in either of the turn signal control circuits (left turn or right turn).
Additional Chimes
The following chimes will turn on when an associated indicator turns on in the instrument panel cluster (IPC):
• Low Fuel Indicator
• Low Oil Pressure Indicator
• OVER SPEED indicator
For a description of an indicator refer to Indicator/Warning Message Description and Operation.
For diagnosis of an indicator refer to Symptoms - Instrument Panel, Gages and Console.

Your have a problem with one of these systems , your best bet , take it to a qualified repair shop. They would do diagnostic's to find the problem .

Do any indicators illuminate when the chime sounds?
Go to Symptoms - Instrument Panel, Gages and Console
Go to Step 3
3
Turn the ignition OFF.
Remove the key from the ignition.
Turn the headlamp switch to OFF.
Close all of the vehicle doors.
Open one or more of the vehicle doors
Does the chime sound at all times regardless if the doors are open or closed?
Go to Step 15
Go to Step 4
4
Close all of the vehicle doors.
With a scan tool, observe the LF Door Ajar Switch parameter and the RF Door Ajar Switch parameter in the instrument panel cluster BCM/Inputs data list.
Do all of the Door Ajar Switch parameters display Door Closed?

With a scan tool, observe the Left Rear Door Ajar Sw. parameter and the Right Rear Door Sw. parameter in the instrument panel cluster BCM/Inputs data list.
Do all of the Rear Door Switch parameters display Closed?

This is a 21 step diagnostic trouble shooting flow chart .

NOTE: The generic electronic module/central timer module (GEM/CTM) is connected to the ISO link and is diagnosed through the data link connector (DLC) with a scan tool.
There are five warning chimes and lamps the GEM/CTM controls: headlamps operating without the ignition on (chime only), door ajar (chime and lamp), key-in-ignition (chime only), driver seat belt warning (chime and lamp), and message center.


Audible Warnings Warning Condition Lamp Duration Audible Duration Audible Output Safety Belt 60 Seconds 6 Seconds Chime, 60/Min. 740 Hz Key In Ignition - Continuous Chime, 120/Min. 740 Hz Headlamps On - Continuous Chime, 240/Min. 740 Hz Door Ajar Continuous 1 Second Single Tone, 1000 Hz Message Center (Tone Request) - 1 Second Single Tone, 1000 Hz
When the is turned to RUN or START from OFF or ACC, the chime will sound for six seconds or until the safety belt is buckled.
The will turn on for 60 seconds or until the driver safety belt is fastened. Both the chime and the lamp turn on at the same time.
If the safety belt is buckled before the ignition switch is turned to RUN or START, from OFF or ACC, neither the chime nor lamp will be activated.
Anytime the is changed to RUN from START while the safety belt warning indicator is activated, the safety belt warning indicator will remain activated until either the timing sequence has been executed or the safety belt is buckled.
When at least one door is ajar and the key is in the RUN position, the door ajar cluster lamp is activated and remains on until all doors are closed. A warning tone will sound once any time the key is in the ignition switch and any door is open.
The key-in-ignition warning will sound when the driver side is opened with the key in the and the in OFF or ACC. It will continue to sound until the key is removed, the is closed, or the is in RUN or START.
The headlamp-on reminder chime will be activated if the headlamps or parking lamps are on, the key is not in the ignition switch, and the driver door is opened.
A single tone will be provided each time a ground signal is received from the message center. A one-second tone will sound for warning indications from the message center. A 100 milli-second tone will sound for each message center button depressed.

Key in ignition chime ?
Saturn L-Series ?????? L what ? L100 ,L200 , L300
Audible Warnings
The body control module (BCM) will provide an audible warning for various events relating to the vehicle. The functions that provide an audible warning are the key in ignition, the lights on, the occupant restraint (seat belt) and the park brake.
Key in the Ignition Audible Chime
The BCM will provide an audible chime if the key is left in the ignition in either OFF or the accessory position with the driver's door open. The chime rate for this is 150 chimes per minute. The key in the ignition chime has a higher priority than the lights on chime.
Lights on Audible Chime
The BCM will provide an audible chime if the park lights are on and the driver's door is open. In addition, 1 or 2 other conditions exist. The key is out of the ignition, or the key is in the ignition in the OFF position, and has not been turned. The BCM will chime at a rate of 200 chimes per minute until the conditions are removed.
Occupant Restraint Audible Chime
The BCM will provide an audible chime based on the class II message received from the SDM if the driver's seat belt is not buckled when the ignition is turned to run or crank. The chime rate for this feature is 50 chimes per minute.
Park Brake Audible Chime
The BCM will provide an audible chime for park brake when the park brake is applied and the vehicle speed is higher that 3 mph. The chime rate for this feature is 200 chimes per minute.

You can't disconnect anything . The problem has to be fixed .

The BCM will provide an audible chime for park brake when the park brake is applied and the vehicle speed is higher that 3 mph. The chime rate for this feature is 200 chimes per minute.
Audible Warnings (Chime)
Chime Rate Per Minute
Key in the Ignition Audible Chime
150
Lights on Audible Chime
200
Occupant (Seat Belt) Restraint Audible Chime
50
Park Brake Audible Chime
200
Content Theft Pre-Alarm Audible Chime
200
Telltale Chimes
The body control module (BCM) will provide 4 chimes for each telltale activation per ignition cycle, at a rate of 150 or 200 chimes per minute. The chimes are limited to 1 chime event per telltale for each ignition cycle. The power up telltale function will not activate the telltale chime function. If the telltale becomes active during a cycle then the chime will sound.
The following telltales will activate the chime function:
• Low Washer Fluid
• Change Oil Soon Indicator
• Brake Fluid Low
• Oil Pressure Low
• Coolant Temperature Hot
• Low Coolant Level
• Low Fuel Level
• Antilock Brake System
• Volts Light (Generator)
• Supplemental Inflatable Restraint System (Air Bag)
• Trunk or Hatch Ajar

Having the vehicle checked for DTC'S - diagnostic trouble code's in the BCM would be the first step in the diagnostic procedure .
DTC B2965
When the ignition key is seated in the ignition switch, the key-in-ignition switch contacts the close, grounding circuit 80. The body control module (BCM) monitors the state of the key-in-ignition switch, the state of the ignition switch (OFF, ACC, or RUN) and the driver's door to control the key-in-ignition chime.

It is normal operation for the odometer to light up with the driver's door being opened. It is not normal for the chime to go off. The "Key-In Switch" in your steering column is malfunctioning. The body control computer "Thinks" the key is in the ignition when it is not. It usually requires a new key tumbler assembly to fix this.

Retrain the fobs following the procedure in your owner's manual:


Once entering this procedure, programming the first transmitter erases all previous transmitters. You must program all transmitters.

Program one vehicle at a time. One transmitter could be programmed into other vehicles that are also on
program mode in close proximity.
1. Sit in the driver's seat.
2. Remove the ignition key from the ignition lock cylinder.
3. Close all vehicle doors.
4. Press and hold unlock on the door lock switch.
5. While holding the door lock switch in the unlock position, insert and remove the ignition key twice. Do
not rotate the ignition lock cylinder.
6. Insert the ignition key a third time leaving the key in the ignition lock cylinder. The ignition must remain
in the LOCK position for the rest of the procedure.
7. Release the door lock switch. You will hear three chimes. This chime feedback indicates that transmitter
programming is active.
8. Press and hold the lock and unlock buttons at the same time on the keyless entry transmitter for 12
seconds. You will hear two chimes. This chime feedback indicates that the transmitter has been
programmed successfully.
9. Repeat step 8 for the rest of the transmitters (for a total of 4 transmitters).
10. Exit this procedure by removing the ignition key from the ignition lock cylinder.

Section 13-09: Gauges, Warning Devices — Chime System 1996 Windstar Workshop Manual
DESCRIPTION AND OPERATION Warning Chime Systems Three distinct sounds are used to indicate the area of concern. Seat belt warning generates 1 chime per second
key-in-ignition generates 2 chimes per second
headlight on generates 4 chimes per second.
Key-In-Ignition Warning The key-in-ignition warning will sound when the driver's front door (20125) is opened with the key in the ignition switch. It will continue to sound until the key is removed, the front door is closed, the ignition switch is turned to START or the vehicle speed is over 24 kmh (15 mph).

Safety Belt Warning When the ignition switch is turned to RUN or START from OFF or ACC, the chime will sound for six seconds or until the safety belt is buckled.

• The safety belt warning indicator lamp (10C859) will activate for 60 seconds or until the driver's safety belt is fastened. Both the chime and lamp activate at the same time.

Anytime the ignition switch is released from START and returns to RUN while the safety belt lamp or chime is activated, the safety belt warning lamp or chime will remain activated until either the timing sequence has been executed, or the safety belt is buckled.

Headlamp Switch On Warning The chime will be activated if the headlamps or parking lamps are ON, the key is not in the ignition and the driver's door is opened. The chime will sound at a rate of four per second.

Warning Chime Systems The generic electronic module (GEM) will sound the following warnings:

• key-in-ignition warning
• safety belt warning
• headlamp switch ON warning
• door ajar warning

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Headlamp Switch On Warning The generic electronic module (GEM) sounds when the driver door is opened while the headlamp switch (11654) is in the PARK or HEAD position, and continues to sound at a rate of 4 chimes per second until the headlamp switch is turned off or the driver door is closed.

• When the headlamp switch is in the PARK or HEAD position, power is supplied through Circuit 293 (O/R) to the GEM.
• When the driver door is open, the driver door open warning lamp switch (10B997)is closed and ground is supplied to the generic electronic module (GEM).

A delayed headlamps on repetitive chime will sound for a period of three seconds beginning twenty seconds after the key is turned to OFF, LOCK or ACC, the headlamp switch is in the PARK or HEAD position, and the driver door is closed. Delayed headlamps on is terminated by opening and closing the driver door again.
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Door/Liftgate Ajar Warning The door/liftgate ajar warning functions as follows:

• The generic electronic module (GEM) sounds a warning tone for one second when any vehicle door, liftgate window or the liftgate is ajar without any other door already open and the ignition switch is in the RUN position.
• Power is then supplied through Circuit 627 (BK/O) to the door ajar warning indicator lamp which remains illuminated until all doors are closed or the ignition switch is no longer in the RUN position.

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Key-in-Ignition Warning The generic electronic module (GEM) sounds when the LH front door is opened with the key in the ignition switch and continues to sound until the key is removed or the door is closed.

• When the key is in the ignition switch, the ignition key warning switch terminal and wire (11A128) is closed and ground is supplied through Circuit 158 (BK/PK) to the generic electronic module (GEM).
• When the LH front door is open, the LH front door open warning lamp switch (10B997) closes and ground is supplied through Circuit 57 (BK) to the generic electronic module (GEM).

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Safety Belt Warning When the ignition switch is turned to RUN or START from the OFF or ACC position and remains in RUN or START, the chime will sound for six seconds or until the safety belt is buckled. The safety belt warning indicator lamp (10C859) will turn on for 60 seconds, or until the LH front safety belt is fastened. Both the chime and lamp activate at the same time. If the safety belt is buckled when the ignition key is turned to RUN or START from OFF or ACC, the lamp will prove out for two seconds, but chime will not be activated. Any time the key is changed to RUN from START while the safety belt warning indicator is activated, the safety belt warning indicator will remain activated until either the timing sequence has been executed, or the safety belt is buckled.

The key switch usually breaks or wears out and makes contact. It is on the lock cylinder. This would cause the chime to not come on, or stay on. Sounds like the switch is bad.

1997 F-150/250 Warning Chime, Key-In-Ignition The key-in-ignition chime warns that the key is still in the ignition when the driver's door is ajar. The key-in-ignition warning chime will sound when:

• GEM: Driver door is ajar, the ignition switch is in the OFF/LOCK or ACC position and the keys are in the ignition.
• CTM: Driver door is ajar, the ignition switch is in the OFF/LOCK, ACC or RUN position and the keys are in the ignition.

The key-in-ignition warning chime will stop sounding when:

• Driver door is closed.
• Key is removed from the ignition.
• GEM: Ignition switch is switched to the RUN position.

The key-in-ignition warning chime inputs are:

• Ignition switch RUN position (battery potential on both RUN and RUN/ACC inputs).
• Ignition switch ACC position (battery potential on RUN/ACC input only).
• Key-in-ignition switch (grounded with key-in-ignition, open circuit with key-out).
• LH door ajar switch (open circuit with door closed, grounded with door ajar).

This is a pic for a 98, could not find one for 97 which may be the older style which is more prone to break.

SYSTEM OPERATION

System warning lamp located on top of the instrument panel
Press UNLOCK button once , turns on interior lamps and deactivates alarm (if fitted)
Press UNLOCK button twice w/in 5 seconds unlocks all doors, turns on interior lamp and deactivates alarm (if fitted)
To activate alarm system (if fitted):
Switch ignition OFF
Open driver's door
Press Lock button on remote
Close Driver door
warning lamp will flash when system is activated
Press remote power sliding door button will open sliding door (if fitted) only if unlocked first
Pressing remote panic button:
*Sounds horn intermittently
*Flashes exterior lamps
*Turns on interior lamps
**Pressing panic for the second time OR switching ignition ON deactivates alarm.


PROGRAMMING
Remobe key from ignition switch
Remove the "BCM PRGRM" fuse, from the fuse box
Close all doors
Turn ignition switch to the ACC position
With in 1 SECOND , turn the ignition switch to OFF then back to 'ACC'
(really, with in one second)
Open and close the driver's door. A chime will sound, indicating programming mode activated
Press and HOLD LOCK and UNLOCK SIMULTANEOUSLY untill a chime sounds 2 times ( about 15 seconds)
Repeat this exact procedure for all remotes to be programmed
Remove key from ignition
Refit 'BCM PRGRM' fuse.

4 REMOTES MAXIMUM CAN BE PROGRAMMED