I had a similar problem, and a variation on the quilt-method worked for me.
It appeared that the screen stopped working so I tried connecting it to an external monitor and got nothing. No startup chime, resetting PRam didn't seem to work, etc. HDD spun up, power light came on and if you close the lid it pulsed.
I wrapped the bottom half of the machine in aluminum foil (not the screen), taking care to block the fan exhaust.
I plugged in power and a USB mouse so I could wake it up while closed, and closed the lid.
I clicked the mouse & made sure the machine was on (solid power light).
Then I wrapped the whole mess in a microfiber blanket to keep it very warm, and cooked for 2-3 hours.
When I came back, the machine was pretty hot. I let it cool to room temperature, turned it on, and it worked perfectly fine.
I was completely blown away that this worked, and figured I should share my experience. Now I'll take it into a genius bar to see if they will agree to replace the motherboard as a courtesy. They've done this for me on other systems in the past.