it probably isnt smart, but i will say i have thrown a stock cooler with no mounting hardware, and used paste left on the cooler to quickly do a test boot before, and made sure i held it down, and it was fine, and i would think the same as for no paste, but when i mean quick, barely a minute, but temps were fine as paste was still slightly there and the heatsink wasnt hot. With no either dont chance it, or hold it down securely, and get to the bios post splash screen and thats prob as far as i would go. With no cooler or paste it will most likely end badly....
The paste is there for efficient transfer of the heat between the cooler and cpu headspreader, direct contact should be enough in theory for a quick bootup. Your hardware not mine, if it cant wait until tomorrow for u to go buy some paste then is it worth it?