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.