Adding my two cents... First, do you hear the Apple Bong? Could be something misconfigured and needs a reset (NVRAM, SMC, PRAM). While the possibility is a overheating system, you can test that theory simply by holding the option key down and leaving the system idle on the Startup Manager. If the screen stays on indefinitely and the system has built-up heat coming of it, then it's probably not heat related. The LCD and power supply are probably OK at this point. The next issue to try and diagnose would then be something related to the hard drive. Data corruption due to a failing hard drive would drag the boot speed down and possibly cause a shutdown.
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