Continuity mistake: During the big song near the end, there is a birds eye view of everybody outside dancing and singing, and Krusty, Chief Wiggum and Skinner are there outside with everyone. But in the very next shot, they all appear out one of the windows, singing a line to the song.
Diatribe of a Mad Housewife - S15-E10
Continuity mistake: When Marge is typing her novel on the computer, in one close-up shot of her we see there is a pencil on one side of the table and a pot of flowers on the other. Yet when the shot changes to further away we can see that the pencil has been replaced by a napki, a pot with two pencils has appeared, and the flowers have moved to this side of the table as well. (00:06:20)
Much Apu About Nothing - S7-E23
Continuity mistake: When Homer swings on the electric cable when attempting to get to his car, the cable has just appeared from nowhere because, the previous 'birds eye view' shot showed no cable.
Much Apu About Nothing - S7-E23
Continuity mistake: When Ned is struggling to get in the house at the beginning the door knob is light brown, then dark brown in another shot.
Simpsons Spin-off Showcase - S8-E24
Continuity mistake: In the beaver sketch, "Lisa" is holding Maggie in front of her to show everyone her first tooth. In the next shot, Maggie is instantly to her side.
Continuity mistake: After Apu & Homer find Jasper in the freezer. Apu removes a note from his hand. Minutes later Dr. Nick walks in the door you see Jasper still holding the note then disappears again when Dr. Nick opens the freezer door.
Milhouse Doesn't Live Here Anymore - S15-E12
Continuity mistake: When Bart and Milhouse arrive at Milhouse's home, there is a red car in the driveway. When Bart screams, it has gone. (00:05:10)
Marge vs. Singles, Seniors, Childless Couples and Teens, and Gays - S15-E8
Continuity mistake: When Homer sneaks into Maggie's room to smash the CD player, there is a yellow box with musical notes painted on it. The notes are different in each shot. (00:02:30)
Continuity mistake: When Marge's sisters start smoking in the flashback, there is only one ledge on the wall behind them. In the next shot there are three ledges and a clock. (00:04:05)
Burns Verkaufen der Kraftwerk - S3-E11
Continuity mistake: Homer drives his normal car to the plant's parking lot. When he and Carl meet, his car has changed to being blue. (00:07:00)
Other mistake: When Bart and Homer do a test drive on their race car, it is daytime, when supposed to be night-time. (00:11:50)
Lisa's Pony - S3-E8
Continuity mistake: During Lisa's performance, In one shot of the audience we see, from left to right: Ned Flanders, Maude Flanders and some lady. The rest of the talent show showed us that Jasper was sitting where the lady was. (00:05:45)
Continuity mistake: When Lisa first discovers Jebadiah's confession, she is holding the fyffe and wearing his racoon hat, the screen fades to black (I'm guessing for the ad break) but when it comes back Lisa is standing in exactly the same position, still reading the confession, still holding the fyffe, but no longer wearing the hat.
Continuity mistake: There are two problems with Lisa's Indian burns. When we first see them, she has 4 burns. Then, while she is showing them to the bullies, there are 3, then 4 again. Then, in the very next scene when she is walking with Nelson in the schoolyard, they are gone altogether.
Itchy and Scratchy: The Movie - S4-E6
Continuity mistake: When there is a queue to see the movie, it stretches right back to the Simpson house, and we see Bart looking through the window. However, when Homer talks to Bart whilst sitting on the window, the queue has disappeared. (00:16:45)
The Simpsons Halloween Special: Tree House of Horror - S2-E3
Continuity mistake: When the Simpsons are about to eat on the spaceship, in one shot just before they eat there are 3 candles on the table, and when they all sit down they're gone. (00:10:40)
The City of New York vs. Homer Simpson - S9-E1
Continuity mistake: Just before Duff man arrives, Barney has loads of sweat on his shirt. But when Duff man pushes the big jug of beer towards him in the next shot, the sweat is gone.
The Fat and the Furriest - S15-E5
Factual error: Homer creates a giant sugar ball by making a normal sized cotton candy on a paper cone, and then coating it with liquid caramel. The cotton candy holds its shape and Homer continues to expand it with more cotton candy and caramel. However, cotton candy dissolves very quickly and tears apart easily. Between the weight and moisture content of the caramel, the original cotton candy would have immediately melted and/or collapsed.
Skinner's Sense of Snow - S12-E8
Audio problem: When Principal Skinner is watching the kids throw the books in the fire, someone goes to throw Huckleberry Finn into the fire. He says "Not Huck Finn. I spent hours crossing out the sass-back." While he says this, his lips don't sync with the words.
Continuity mistake: When we see Skinner's car before the newspapers fall on him, it is a light-brownish color. When the police arrive, it is the same color as the cop cars. (00:19:10)
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