Much Apu About Nothing - S7-E23
Continuity mistake: When the mob come back to Quimby to complain about the 'Bear Tax' he opens the left door inwards.In a following shot both doors open outwards.
Much Apu About Nothing - S7-E23
Continuity mistake: When Apu awakes, he has a closed note pad, and the table looks small. In the next high-angle shot, Apu's book is open and face down, with paper on top of it and a pen beside it. The table is also bigger, with Bart sitting there with his breakfast things too.
Much Apu About Nothing - S7-E23
Continuity mistake: When Nelson pulls back Uta, he drops a book. In the next shot, the book has gone.
Much Apu About Nothing - S7-E23
Continuity mistake: When Apu and Lisa go to the 'Immigration Centre', they pass Dr Hibbert on the right, and he has yellow skin. In the next shot, his skin returns to its proper brown colour.
Much Apu About Nothing - S7-E23
Continuity mistake: When Bart pulls out Apu's punch cards, there are many sitting on top of the box. When Apu picks the box up and places it in the bin, there are only a few cards sitting on top.
Much Apu About Nothing - S7-E23
Continuity mistake: Homer comes to the Kwik-E-Mart with the new 'American Apu' and sees him reading a magazine. When Apu walks over to Homer, his magazine disappears.
Much Apu About Nothing - S7-E23
Continuity mistake: In the early scene where the Simpsons are gathered around the table to eat, you can see that they are eating slices of pie. However, on some of the long shots, the food on their plates changes and looks more like mashed potato and vegetables. Also, in the same scene, in one shot of Lisa, you can see Bart's hand over her plate, but when the angle changes, it is behind her plate.
Much Apu About Nothing - S7-E23
Continuity mistake: In the final high-up shot of the Simpson's garden with everyone chanting 'No on 24'; Lisa has disappeared from next to Apu, and Ned Flanders' glasses have vanished.
Much Apu About Nothing - S7-E23
Continuity mistake: When Willy is threatening the children and Skinner it shows that Lisa has her hands on her sides and she is holding nothing. In the following shots, she has a blue book held against her chest.
Much Apu About Nothing - S7-E23
Continuity mistake: When the mob return to Quimby his assistant's clipboard changes colour between shots.It goes from light brown to burgundy then back to light brown again.
Much Apu About Nothing - S7-E23
Continuity mistake: During the party for Apu, Dr. Nick is first seen not wearing a coat, then he has his usual white coat, then he has a brown coat, and then the white coat again.
Continuity mistake: After Homer dodges the cannonball near the end of the episode, the blue platform that he stood on goes further away from the edge of the stage between shots, and when Homer hugs Marge the platform has disappeared and Peter Frampton's cooler has become closer to Homer.
Continuity mistake: When Homer is on stage in Springfield and preparing to get hit by the cannonball, the speaker-tower, behind and slightly to the left of him, completely disappears in the wide shot right before Homer jumps off stage.
Continuity mistake: After Homer is shot with the cannon for the first time, the cannonball disappears from behind him.
Continuity mistake: When Homer is taking the kids to school for the second time the view behind him doesn't look further away while driving.
Continuity mistake: In the scene where Homer is on stage in Springfield, he's wearing shorts. When he jumps off the stage after dodging the cannon ball, he's wearing pants.
Continuity mistake: In the scene where Homer dodges the cannon ball, he is wearing goggles and a helmet. But when Homer jumps down from the platform and hugs Marge, his helmet and goggles have disappeared.
Continuity mistake: The X-rays of Homer's intestine change sides between shots when he is at the vet.
Continuity mistake: When Lisa meets the other kids for the first time under the boardwalk, her tinted shades disappear in the very last shot.
Continuity mistake: When Marge is cleaning the dishes on the morning after Bart reveals Lisa as a nerd, Marge is seen in close-up with a white necklace. In next shot of Bart and Lisa, as well as the rest of the scene, Marge has a red necklace.
Answer: I'm not sure if this is the one you're thinking of, but an episode of "Ray Bradbury Theatre," called "A Sound of Thunder," dealt with a similar matter: a group of hunters travel back in time to hunt dinosaurs, only to find things have changed when they get back because someone stepped on a butterfly.
Xofer