A processor will shut off if it gets too hot, so you may not do any damage.
But, it will get hot so quickly that the test will be useless.
Moreover, many motherboards will not start if it does not detect a cpu fan.
Most any cooler you buy will come with thermal paste.
If you happen to have a cooler, but no paste, I suppose you could use it.
I have read about some using smooth peanut butter or toothpaste which works temporarily.
------------------->Bottom line, wait.