Revealing mistakes: During the Fonfon Rubok ceremony Amy's mom blinks, as her eyes shut, the lines composing the animation disappear and she is left with no eyes.
Continuity: When Kif says hello to Fry on the missile he switches back and forth from being on the left and right sides of its fin.
Continuity: The remnants of Fry's pink note disappear and reappear on Bender's sword during the climactic fight with Yivo.
Continuity: Hedonismbot's mouth and eyes keep changing throughout the film. His mouth is either shown with just one row of teeth or one row of teeth and two lines creating a "mouth" like Bender. Also his pupils change from being round to being square like most robots.
Revealing mistakes: When Fry and Colleen are lying in bed, post-coitus, his hand is around her shoulder but it is the same colour of Colleen's skin. (And not his own flesh tone - it vaguely looks as if she has three hands.)
Continuity: During the first 'Deathball match', the booth window is smashed and Wernstrom gets knocked onto the court, however in the next overview shot, the window is intact and he is still stood behind it.
Continuity: When Stephen Hawking is talking in the science conference, the microphone disappears in the last shot.