Continuity mistake: Before Homer sees his vision of Caesar Romero the day sign next to him says 9. After the Duff executives see him talking to nobody they decide to take him away and when they get down there the sign now says 12 with the president of Duff confirming it in his speech. Doubtful it took them 3 days to get down to him and not much reason for Homer to take 3 days off the sign seeing as he was almost passing out from the lack of food.
Continuity mistake: Just before Marge tries to sell Skinner the tie rack, she is writing on something. She then puts her hands down by her waist, and then brings one up to turn it on and both up to pull the motor out. The point is that the pencil and the item she had in her hands have simply vanished, and we never saw her put them down either.
Continuity mistake: When Professor Frink is explaining where Homer is, in "Homer3", Ned Flanders is leaning behind the couch between Bart and Marge, and Chief Wiggum is standing up. When Homer complains, Ned is between Marge and Chief Wiggum, who is now sitting down on the couch.
Continuity mistake: When Bart crashes at the door of the Legitimate Businessman's Social Club, his head is facing the door. You can only see this for several frames. In the next shot, his head is facing the sky. (00:05:25)