Continuity mistake: When we see the professor's clone inside the tube there is a panel attached to a metal post going down to the bottom of the tube. When the professor hits the button to empty the tube of fluid it cuts to a close-up of the clone's feet and the metal post is suddenly gone.
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?