Thermal paste increases the efficiency of the heat transfer between your cooling fan and CPU. It's not optimal to run without it, but it has no impact on power.
The symptom you described (a failure to POST) *could* be caused by your removal of the thermal compound, but is more likely to be related to some other issue.
Removing the thermal compound would cause the failure you're describing IF the chip was instantaneously overheating and tripping the motherboard's thermal threshold protection - OR - if you've permanently damaged the chip via overheating.
I believe it is more likely one of the following:
1. You cleaned your chip which likely means you've removed the chip, wiped it down, then replaced it. If not seated securely, the CPU will not be locked in appropriately and the system will fail to POST***
2. You likely were working to clean the whole of the inside of your computer which I would imagine would relate to everything, possibly including the RAM. Check to make sure that ALL of the RAM sticks are seated securely (press down firmly but gently on both ends of each stick and listen for a slight 'click' sound). If you hear that, you've likely found the problem.
3. You have a 20, 20+4, or 24 pin power connector to your motherboard. If you were in your case and unplugged this (because the cable is stiff and awkward to work around), and you didn't plug it back in all the way (so the plastic clip has the plastic harness securely locked to the motherboard), then it's possible that not enough juice is making its way to the board which will cause a fail to POST.
4. You may have been cleaning a video card as well. It is possible that the PCI card is unseated properly (or if it's new, your power supply doesn't have the capacity for it) and the card will draw too much power, thus causing a shortage on the motherboard, and it's possible that you'd get what you're seeing. Unplug your video card (and any other PCI cards you have connected) and give it a shot.
5. If you've been really messing around inside your case, and you've unplugged some of those obnoxiously small wires (typically down by the bottom of the case), and you plugged them back in from memory, then you've cross wired the cabling that connects your case's power button to the motherboard (power can become reset, etc.). Consult your motherboard's user manual for explicit wiring instructions
***NOTE***
I've spent hours of my life troubleshooting the problem you're facing. Almost every time, I hadn't fully seated the CPU properly. Check this first, then work your way through the rest.
If you're still having trouble, it could be a component failure, or some other small config adjustment and you'll have to take that to the forums.
Best,
TW