You could give it a chimney cover type of thing, like this:With a fire or stove, rain and snow would be absorbed into vapor in the hot air and/or pushed out. Without that - wouldn't it come in?
Though I imagine the little roof thing would need to have massive overhangs to keep snow from blowing in from the sides. The teepee would need some serious modifications to the design for this to work... A fun challenge for somebody with a lot of time on their hands, but I'm guessing overall less practical than a standard coop.