Homer Alone - S3-E15
Continuity mistake: In the scene Marge goes crazy in the car, Homer is let through, and he passes under 'distressed mother' police tape, but in the next shot, the tape has disappeared. (00:06:40)
Tree House of Horror VI - S7-E6
Continuity mistake: When the advertisement of Kent Brockman comes to life he is a thin cut out from a bill board, however the next time you see him he is in the shape of Kent Brockman and no longer thin.
Marge vs. Singles, Seniors, Childless Couples and Teens, and Gays - S15-E8
Continuity mistake: When we see The Town at the meeting, Smithers is sitting in front of Selma. In the next shot, though, Smithers is suddenly on stage holding the 'SSCCATAGAPP' sign and Lenny is sitting in front of Selma. (00:08:50)
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.
Alone Again Natura-Diddly - S11-E14
Continuity mistake: Homer's watch disappears when he is getting his car fixed.
Continuity mistake: When Homer sees that Bart and Lisa aren't listening to his story, he looks around the living room while sitting on the couch. He appears beside the window in the next shot. (00:03:30)
Continuity mistake: In the flashback, at the Aztec Theater, the person in front of Marge and Homer changes. (00:04:40)
Springfield Connection - S6-E23
Continuity mistake: When Marge is shopping, there is a blue baby seat in her trolley. In a following shot of her running with the trolley and it going into the cheese display, the seat has gone.
Continuity mistake: At the institute, Dave puts his hand on his son twice, and they also swap positions between shots. (00:10:00)
Continuity mistake: At the Institute, the clipboard Dave is holding repeatedly switches arms between shots. It also disappears entirely. (00:09:35 - 00:10:35)
Continuity mistake: When the family are walking through the car park after the visit to the Knowledgeium they walk by a purple car and there is a quick visual of a blue car in the bottom left hand corner of the screen.They immediately stop and they are now next to a pink and brown car.
Continuity mistake: When Homer is in space, when he is about to free the ants, the crisps that he opened earlier have disappeared from the background. This type of mistake happens again when both the crisps and the ants disappear and reappear between shots.
Continuity mistake: Near the end of the episode, when Skinner shouts at Ralph to shut up, the sky outside the bus changes from nighttime to daytime between shots. (00:14:15)
Continuity mistake: After Sideshow Bob finishes reading his book and puts it on the table, the pages of the book change from yellow to red between shots.
Continuity mistake: When Bart is talking to Jessica after she stole the collection plate money, she is far away from Bart. But when Bart says "but you're not very nice", she is right next to him.
Continuity mistake: As Bart is forced to show Marge the item (picture frame) he is hiding, he pulls it out of his jacket and he hands it to Marge. As he pulls it out you can see there is nothing in the frame or on the border of the picture frame. After Marge's reaction to the photo we see a close up of the frame and there is now a picture in the frame and some colorful decorations around the frame. Whilst the frame could have been the backside when he pulled it out he never turned it over when handing it to her, and Marge never turned it around when looking at it.
Tree House of Horror VI - S7-E6
Continuity mistake: During the 'Nightmare on Evergreen Terrace' segment, when Bart is running from Willie, he plans to sink Willie in the sandbox. In the first view of the sandbox it is isolated on a field, yet when Willie is sinking in the sand there is a jungle-gym beside the sandbox. (00:11:35)
Continuity mistake: At the garbage man debate, when Homer says 'Then you should have been early.', we see a shot of Lenny laughing. To the right of him, there is an old woman. In the next shot of Chief Wiggum, the same old woman is sitting to the right of him. In the later shot of Wiggum, Miss Hoover is sitting to the right of him and the old woman has disappeared.
Moaning Lisa - S1-E6
Continuity mistake: When Bart and Homer first play the video boxing, the window behind them shows that it's broad daylight outside. When Homer blames his loss on Lisa's Sax, it cuts to Lisa's room and it's night all of a sudden.
Secrets of a Successful Marriage - S5-E22
Continuity mistake: When Homer wipes the tobacco on his forehead in his new class, the tobacco just disappears on his hand/sleeve.
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