5 minutes will be an eternity. I've turned on a PC without a HS just to see if it would post. In the time it took to turn on and post, I kept my finger on the exposed CPU. It was already hot. not quite hot enough to burn, but that was only after about 10 seconds. I would wager that it would either turn off or die in less than 30 s.
About the thermal paste, it isn't technically needed, but it IS very useful. If a heat sink is clamped onto a cpu right, it will conduct enough heat away for your processor to work fine. Thermal paste (basically) adds more contact area so that more heat can be transferred.
It was a bet with my friend. He laughed at me because he said computers don't need thermal paste (by heatsink thats what i mean') I mean, most do.. but I guess some older ones don't..
So you actually mean with a heatsink and fan but without thermal compound?
It would post. It would run. It would just run hot to very hot depending on the CPU. I did a bunch of experiments with and old Athlon 64 3200+. It ran 10*C hotter without compound. I also removed the heatsink, slowly, during operation. It auto shut down somewhere above 115*C.