Deliberate mistake: After Wolverine arrives at 3 Mile Island he approaches through a thick fog. Above him are a series of conveyor belts that are used at a coal power plant, not a nuclear plant such as 3 Mile Island. The belts are unique to coal mines and coal power plants, and would not be found at a nuclear power plant. The filmmakers obviously couldn't film on the site of a nuclear plant, so filmed on a coal plant to fill in.
Deliberate mistake: After being arrested, Breaker, who is Northern African, a region where French is the second if not first native language and speaks with a French accent throughout the movie, pleads with the French authorities in English. (01:10:30)
Deliberate mistake: When Selfridge says "Pull the plug" after Hometree is destroyed, two soldiers, one male and one female, go to shut off Jake and Grace's avatar links. The female soldier goes straight to Grace's link and terminates it, but the male soldier gets delayed (punched by Norm) before he can terminate Jake's link. This is deliberately shown out of sequence - we see Jake's link terminated before Grace's. The result is that we see Grace's avatar fall unconscious when Jake's link is terminated, and Jake's avatar fall unconscious when Grace's link is terminated.
Deliberate mistake: Though the film takes place over the course of several weeks, at night there is always a full moon.
Deliberate mistake: In the scene where we see Bella in her room looking out the window, the months are passing, but she is always wearing the same clothes. (00:28:30)
Deliberate mistake: When Bender wins the poker tournament with five kings, Leo Wong holds up five fingers (on one hand). But all the characters in Futurama only have four fingers on each hand. (Confirmed on DVD commentary).
Deliberate mistake: After the SAD unit shoots down the command module, Christopher falls to his knees in sadness. In the next shot, the mercenary supposed to be pointing his gun at his head is looking and pointing as if Christopher was still standing, as revealed when he actually gets back up again.
Deliberate mistake: It is totally unlikely that a museum as careful and important as the Smithsonian would leave the crates un-nailed, which could make the pieces inside get damaged. Of course if the crates weren't open, none of the figures could come up to life.
Deliberate mistake: When Sam is watching the security footage of his predecessors being disintegrated in the chamber, the disintegration happens the instant the coffin door is shut. However when he goes in to test it himself there is a ~60 second wait as a congratulatory video plays before the disintegration occurs. (01:00:00)
Deliberate mistake: A branding iron loses its bright orange glow relatively fast when it's out of the fire. When the police enters the room it is still very bright orange even though it has been on the floor for some time.
Deliberate mistake: Helena asks Justine over to watch Atonement, but she instead goes on her 'date' with Alexis. When the scene cuts to Helena and Emily watching the film, it's actually The Duchess.
Deliberate mistake: A flare thrown on top of a snow-covered mound of logs/twigs is not able to engulf the area as fast as it does. (01:13:13)
Deliberate mistake: In the link between the movie reel and nightclub scenes a train goes past. While the sound effect is convincing, the video has been reversed. The headlights on the train give it away, but cars under the bridge are also seen in reverse. (00:27:55)
Deliberate mistake: When Janine smashes the mirror she takes it directly off the wall and raises it above her head, and the reflective side would be facing behind her. The camera immediately cuts to a front view as she's throwing it down and the reflective side is in front. This was probably done for dramatic effect so that the reflective side would hit the ground first.