Um...
The heat sink has to have the thermal grease or a thermal pad which is basicly the same thing. Grease is better. The fan has to be plugged in.
As for hurting the p4 chip, they are pretty forgiving as far as temps go. The built in thermal shut down protects the cpu chip. Sounds like something on the motherboard is shorting out. Or the psu start command is not locking in. Shut everything off and do this test.
Unplug the power plug from the mother board along with all the power plugs inside the computer. Put a jumper between the green wire and a black wire and see if the power supply's fan comes on. Take a fan from the case and plug it in to the power supply. If it spins up then you're good for the most part. Shut it off and put it all back together.
Take a light and look at the header where the led's and switches connect. Make sure all connections are where they should go and that the led polarity is correct. You can consult the owners manual or get the specs off the MOBO website. If you find anything wrong correct it with all the power off. That means the kill switch on the psu or yank the plug out of the wall.
Now I've seen where a scrambled bios can do something like this but i kinda doubt it in this case. However let's give it a go. Power off. Either remove the cmos battery or use the reset jumper. Give it a couple of mins and put it back just like you found it. Power up the psu and attempt to boot.
And although this may sound a bit dumb did you remove the motherboard from the case and put it back in? If you did, did you use the stand offs or mount it directly to the mounting plate?