Continuity mistake: As the power builds up in Bender, his left eye pops out, making him a cyclops, subsequently being eaten by the Cyclops eater. After the monster dies, Bender comes out, and has two eyes in the long shot, but puts his left eye back in the close up.
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Continuity mistake: When Fry and Bender are in the museum and locked in the "Villains" room, Bender breaks the two girders in the window, and then his arms falls off. As he puts his arms back in, you can see behind him that the girders are back in place in the window.
Continuity mistake: When Fry and Leela get to the moon landing site, Leela climbs in the moonlander hatch with the oxygen tank to which Fry's oxygen hose is connected. The hose is visible as it leads in to the hatch. When Leela pops her head out the hatch, the hose is suddenly gone. Then in the next shot, it's back, but lying underneath the hatch.
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.