Other mistake: When the Moon Patrol officers throw Bender out of the airlock onto the moon's surface, they are exposed to hard vacuum, but they aren't wearing space suits. Throughout the episode it is established that the moon's "atmosphere" is a vacuum and protective suits are necessary - when the farmer removes his helmet he begins to asphyxiate instantly.
Other mistake: When Professor Farnsworth was presenting the crude drawing of the Smelloscope to his scientific peers, he wiped the napkin which it was drawn on on his forehead, causing all the ink to smudge. But why isn't there any ink on Professor Farnsworth's head? (00:05:30)