Mitsubishi 65" LCD (1080P, DLP) keeps shutting down until it "warms up". After shutting down the power led flashes yellow for a minute or so, at which time I power it up again. This cycle repeats 1 to 4 times before it stays on. What should I do?

Failing power supply

Those blinks [flashes] are fault indication (Error Codes). From the number of blinks occur at one time switch ON, you can get an idea about the faulty circuit section inside it. This will make repair easy. These are called blinking codes. Make sure about the exact number of blinking of the standby/power LED, by switching On & OFF the set, for at least two times. Count it carefully. If you wish to get some details, check the site linked here. It has details about blinking codes, and fault locating procedure, to many popular brand TVs and other devices.
http://homenol.blogspot.com/
http://electronicshelponline.blogspot.com/
If blinks continue without any pause, the main board is faulty.

I would have the power supply checked for the screen and then have the power button checked it may be faulty.

Try a reset. Start by unplugging for 1 minute, then plug back in & if still won't restart try the following. If none of the following work, you will need a technician.

Initializing the TV in many cases can remedy a problem.
In the 2006 product line Initialization is now a user function.
Pressing the front panel "INPUT' and "FORMAT"
buttons at the same time activates Initialization. When
complete all user settings will have to be set again. The
table in Figure 4-3 list the User Initialization procedures.
Note: Pressing INPUT and FORMAT is the same as
the Initialization in the Option Menu (MENU-2-4-7-0)
Order Name Procedure Results Notes
1 System Reset Press <System Reset>
on Front Panel
Hardware & TV operating
software reset.
POWER/Timer LED flashes for 60
seconds. No customer settings
are changed.
2 User Initialization
Press <INPUT> &
<FORMAT> on Front
Panel at the same time.
All customer Audio &
Video settings are reset to
factory default.
All customer settings will need to
re-configured.
3 Reset System
Defaults
From the remote, press
<MENU> <1-2-3>
<ENTER>
All customer settings
except V-Chip are reset to
factory default.
Netcommand, Channel Memory,
etc., will need to be re-configured.
SERVICER
PROCEDURE 4 Initialization
From the remote, press
<MENU> <2-4-7-0>*
select "INITIAL" press
<ENTER>
All customer settings are
reset to factory default.
All customer settings will need to
re-configured.
MITSUBISHI RESET / INITIALIZATION CHART
CUSTOMER
PROCEDURES
Figure 4-3
Figure 4-4
LED Indications
The front panel LEDs, refer to Figure 4-4, also assist in
determining the cause of a symptom. Figure 4-5 illustrates
normal LED operation. Figure 4-6 shows the
LEDs response when a problem is detected and indicates
the possible cause.
Power/Timer
Indicator
Status
Indicator
Lamp
Indicator Condition
Off Off Off Off (standby)
Blinking green
fast Off Off Initialization (10 - 65 seconds)
Off Off Blinking
green fast Lamp runs for 1 min after PTV off
Green Off Off Power On
Blinking green Off Off Power On Timer is set
NORMAL LED INDICATIONS
Power LED Status LED Lamp LED Condition
High Temperature - check ventilation air flow.
(Temperature sensor is disconnected.)
Off Off Blinking yellow Lamp cover is open.
Lamp failure (Lamp fails to turn on)
Off Off Red (Lamp turns off after lighting)
Lamp is broken
Off Blinking Red Off Fan Stop
Off Red. Off Circuit failure. (Run Error Code Check)

These blinks [flashes] are fault indication (Error Codes). From the number of blinks occur at one time switch ON, you can get an idea about the faulty circuit section inside it. This will make repair easy. These are called blinking codes. Make sure about the exact number of blinking of the standby/power LED, by switching On & OFF the set, for at least two times. Count it carefully. If you wish to get some details, check the site linked here. It has details about blinking codes, and fault locating procedure, to many popular brand TVs and other devices. http://electronicshelponline.blogspot.com/
Blinking codes details to many popular brand TVs are given. Surf the site with patience.
Check the lamp ballast too.

which of the three lights is flashing red?Let me know and I will assist you with the issue.

9 PERCENT IS YOU MAIM BOAR

if it's the lamp led that is orange, turn off the projector for a minute and try it again. if the lamp goes out and the orange led comes on again, replace the lamp and reset the lamp timer.

If you wish to get some details; visit the site linked here. Pull up older posts.
http://electronicshelponline.blogspot.com/
Line faults & possible causes to LCD, DLP & Plasma screen are given with illustrations. You can get an idea about the fault to an extent. Faults to all LCD & Plasma screen are similar, irrespective of its brand and screen size.

It may very well be a connection issue with one of the wire connections as its not giving you a error code. I would double check the wire plug connections