Continuity mistake: There is a weird case of musical tables going on during the Professor's Nobel Prize presentation. When Professor Wernstrom stands and says 'He needs medical attention.' he is sharing a table with an African-American man (who drops his pipe) and a younger man in a green shirt. On the table is an empty retort, an empty glass, one knife and two plates with half eaten meals on them. When he stands and tells the Professor he isn't going to get the Nobel Prize there are three plates with complete meals, a full retort and a full glass and two sets of cutlery. The African-American man has regained his pipe and is at a different table, and a white, bald man has joined them.
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.
Plot hole: Whenever Leela arrives and everyone starts beating up Fry to make him cry out the emperor, it shows Dr. Zoidberg pinching his leg and later, sitting next to the wall, tired out with Bender and Amy. However, if he's there, then what is keeping the Trisolians from coming through the hole in the door that Zoidberg was plugging with his tentacle?