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.