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.
Continuity mistake: In the scene where Leela is telling Fry how even his food is recycled (just after they have seen the video of the origin of the garbage-ball) you can see both Fry's drink can and sandwich on the table. After the closeup of Fry his sandwich has disappeared but in the next full-table shot it's back (albeit moved to Fry's right along with his drink can).