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: 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: 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: 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.