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.