I'm sure it happens. LOL. But this is a 7x11 tile inside. I don't even think one squirrel would stop it enough unless he got all the way down at the stove pipe. (and he likely would have still been there when I opened it)
Birds have a bad habit of hanging around the top of a stone or brick chimney because of the warmth. Had a few, not very often, get overcome by the fumes and whoops, down the chimney they fall into the fire..