Recently I started smelling a funky smell from my computer that smelled like burnt plastic. Assumed it was the heatsink gone bad due to big dust bunnies. Cleaned out the dust bunnies got a big cooler, used good paste but when I removed the old one I put small scratches on the bottom. Didn't bend anything. The new big cooler won't allow for the fan to attach due to the proximity of the PSU (so no fan is plugged where CPU fan is marked). Before I put the new cooler in, I was able to get into my computer to post and get on the web. Started to smell something again and before I could finish researching the web about the problem it shut down. Got the new cooler attached and then it would turn on no beeps, fans (no post) (only on for 2 seconds) then it turns off. Then on again on its own, then off again, then on again. The only way to make it stop is to flip the switch on the PSU or unplug it. Thought it was the PSU but bought a new PSU. First one was a OCZ 700w and this one is a OCZ750w. My mobo is a LGA 775 Gigabyte DDR2 with an Intel 2 Quad Core. 4 GB of ram (2 sticks). Nvidia PCI 2.0 Video Card (forget the number but it has its own heat sink but not power plug for it. Never needed one for it. After putting in new PSU same issue on and off and then on, then off etc... I pulled the memory out and then one by one put it back in. Same issue with or without memory. No beeps. The smell is gone though so that is no longer an issue. The smell disappeared with the new heat sink.
Well the smells probably gone now because turning a computer on and off again causes little to no stress to the component that was most likely overheating. I think your mobo or processor or both have been damaged by this overheating issue. Try turning on the unit with no processor in it. If it stays on powerwise only, its your processor. If it doesn't, then its your mobo.