I grow autos in a tent and when the humidity or the heat gets too much inside the tent (I don't have extractors or anything like that - only a fan), I unzip the damned tent! Yes, of course you can open the closet door! And no, your autos are not going to mind it at all... your yield might be SLIGHTLY less for the light escaping but that's better than the alternative ... That being said, perhaps all you need to do is to put a small fan in there... just get the air flowing around the plant? Unless you have issues of heat/humidity in there, that is what I would do.
Good luck!