Deliberate mistake: The top of the Tipton Hotel is animated. You can tell because in the opening of any episode showing the top of the hotel, there should be 3 visible peaks. You can see 2 peaks and if you look where the middle peak should be, you can see through the top of the hotel to the sky on the other side.
Factual error: In Season 3, "The Arwin That Came To Dinner," in the scene where Arwin is licking his plate clean (that supposedly had food on it), notice that when he does this you can hear that high pitched waxy sound. Oddly enough, if you get a clean plate straight from the dishwasher and rub skin against it, you can hear that same sound. Meaning that there was never any food on it.
Continuity mistake: In the episode "Commerical Break," they start to film the commerical for the first time without London and with Mr. Mosby. London gets upset and sits down on a couch where they are shooting the commerical. She and the couch leave the shot for a second, and London is gone when they show the couch again.