I have a table top cooling fan that requires a 6 volt 400 ma adapter. Is it safe to use a 12 volt 300 ma adapter with it or will it burn the fan up?

Well, I think it should be obvious that something designed to run on 6v will get damaged by running it on 12v.
What exactly will happen and when is hard to say.
I will say I would not do it myself. Those adapters are dirt cheap. Why not get the right one.