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: At the end, when Fry, Leela and Bender are sitting at the table, Fry's legs disappear (and the gray blotch beneath him isn't them - it's the table leg; and it's not hiding them either - his knees should be visible behind it).
Other mistake: When Fry is talking to Leela (After being stitched onto Amy's body), he asks for some privacy from Amy, who puts her fingers in her ears and begins singing a song in Chinese. At one point, when it shows a close-up of Fry's head, you can barely see Amy's head, and the finger disappears and reappears near her nose, then suddenly reappears back in her ear.
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?