Continuity mistake: In the very beginning scene, where Fry and Leela are talking, there is an explosion. When some ceiling falls down there is one big bit and three small bits next to Leela's foot but then a few seconds later there is one big bit and a few small bits all in different places.
Continuity mistake: When Bender and the other obsolete robots are watching a snail crawl across the beach, the snail is shown to be in the center of the screen with the robots standing around. Cut to Bender inserting a CD in his mouth, it then goes back to showing the original angle, but the snail has suddenly disappeared, and not enough time has passed for it to have crawled all the way out of the shot.
Continuity mistake: In one of the first shots at the Freedom Day celebration in front of the Lincoln Memorial, we see a banner, with the text "Freedom Day - Express yourself. I hate my job.", suspended between to poles. Some of the banner can also be seen during the first part of Nixon's speech. But when it gets time to lower "Old Freebie" (the flag) the banner has disappeared and the flag is lowered into the position where the banner hung before. And the banner was tied to the two poles so they can't have raised it.
Continuity mistake: Just as Hermes A is about to shoot the box into the sun, all the crew (A and B) shoot out into the airlock. However, Hermes B doesn't come out of the box and just appears in the pile. This is confirmed in the commentary.
Continuity mistake: On board the Nimbus where Zapp asks for the activation codes for the global defense network, it immediately after cuts to a shot of Kif Saying "Aye Aye sir". In the background of the shot sit two ship crew members, the brown haired one on the left disappears in the following shot.
Continuity mistake: When Bender is being filmed stealing the cigar, we see the "robocop" showing the footage on his screen. In the wide angle you can see he has the siren on top of his head, then in the next shot it's suddenly not there. Then he actually raises it out of his head in the final angle.
Add timeDavid Mercier
Factual error: When Professor Farnsworth visits the Bubbling Geezer to receive youthasizing treatment, the trained professional Heather says "botulism toxin tightens and tones facial muscles." Botulism is the actual poisoning, not the toxin. Botulinum toxin causes muscle paralysis which weakens the muscle, not tightening and toning of the muscles.
Continuity mistake: When Fry, Leela, and Bender make it back to the ship, Fry's hair is messed up. When the bee is unfolding itself, Fry shouts "Look out!" and his hair isn't messed up. Then when he says the bee will have to go through him, his hair is messed up again.