I HAVE A WHIRLPOOL DRYER. TURN THE TIMER SELECT ON. THEN TURN START KNOB TO ON, THEN PRESS THE SAME KNOB IN TO START CYCLE. YOU CAN HERE THE TIMER AND SOUNDS LIKE THE CONTACT IS WORKING, BUT CYCLE WON'T START.

The start contacts in the dryer timer are welded together, the timer will have to be replaced to fix that issue you can order the replacement timer for that model at the link below:
Genuine OEM Whirlpool 53 1810 Timer for Dryer WP53 1810 PS2048133...

A fairly common problem! The official instructions are

Before using your dryer, wipe the dryer drum with a damp cloth to
remove dust from storing and shipping.
1. Load clothes loosely into the dryer and close the door. Do not
pack the dryer. Allow space for clothes to tumble freely.
2. Turn the Cycle Control knob to the recommended cycle for the type of load being dried. Use the Energy Preferred Automatic Setting (,) to dry most heavyweight and medium weight loads. See "Drying, Cycle, and Temperature Tips."
3. Select a temperature setting on models with a Temp/Fabric selector. The cycle descriptions include temperature setting suggestions.
4. Push the START button.

Stopping and Restarting
You can stop your dryer any time during a cycle.
To stop your dryer
Open the dryer door or turn the Cycle Control knob to OFE
NOTE: The Cycle Control knob should point to an Off area when
the dryer is not in use.
To restart your dryer
1. Close the door.
2. Select a new cycle and temperature (if desired).
3. Press START

Extracted from
http://www.manualslib.com/manual/511763/Whirlpool-Lte5243dq.html?page=10#manual

Your timer most likely does need to be replaced. The timer has a small motor that causes it to rotate during the cycle and those can go bad causing the timer to not advance. Unfortunately, you cannot get just that timer motor anymore. Hope this helps!

Hello, we are sorry to hear that you are having this problem with the Dryer. We would recommend having a certified technician out to take a look at the Dryer and give a proper diagnosis. If you would like further assistance you can reach out to us with your name, phone number, address, model/serial number and the date on which you purchased the Dryer to [email protected].

Sounds like you have bad contacts in your timer. By holding the start button and rotating the timer you are just finding a good contact point and allowing timer to send power to the motor. The timer part number is 3979618 and will run you about 90 bucks. hope this helps

_MJ_

sounds like you need to replace your timer, or either unplug the machine, an open the head panel an see if any wires are loose in there, changing the timer is an easy job, you should have no problems doing it ......good luck!

If it is whirlpool--Make sure it has power. Then:Check the door switch. A little lever sticking down through top rightof door opening. If it is not there you will need new doorswitch(sometimes the little lever breaks off). If it is there and doesnot click when you press it in you need a new door switch. if it clicks when you close the door then you have a bad thermal fuse.The thermal fuse is found by taking the back of the dryer off. The fuse is an oblong white plastic piece with 2 wires attached.

Door switch part number 3406109
Thermal fuse part number 3392519

if only rinse cycle work. it mean the cycle timer in no good. you need to replace it. i will tell you the cost of it later.

Check for voltage at the water inlet valve with the timer dial pushed in or in an off position. If no voltage replace the water valve. If you do have voltage you'll have to trace the circuits back from the water valve to the line. One point of demarcation is the water level switch. Pull the violet colored wire and see if it still fills while off. If it does it's likely the timer's bad.

when you turn the select knob, you set up the internal cams of the timer. when you press to start or pull to start, you energize the timer motor (buzz) the timer opens the water solenoids. the diaphragm switch senses the water level and moves the timer to the next step. agitate starts for the timed cycle. when its finished the motor kicks in in opposite direction to drain and pump water out.