Continuity mistake: On the bus at the start there are some lines on Mrs. Krabbapple's microphone. They disappear when she tells Bart to sit down. (00:03:00)
Continuity mistake: At the very start the twins are stood facing each other. Camera cuts to behind and now they are side by side. (00:01:20)
Life on the Fast Lane (a.k.a. Jacques to Be Wild) - S1-E9
Continuity mistake: When Lisa asks "What did you get?" when holding the tray, you can see next to her there is a cup, but in the next shot the cup has changed into a glass of milk.
Life on the Fast Lane (a.k.a. Jacques to Be Wild) - S1-E9
Continuity mistake: When Lisa is speaking to Bart in the kitchen at the start of the episode, Lisa's hands are on the kitchen top. In the next shot, they aren't. (00:00:25)
Continuity mistake: When Skinner is showing Homer the "fake" sick note, the writing is clean and tidy. When we see Skinner comparing the note to the cheque the handwriting is suddenly different. (00:07:15)
Continuity mistake: The writing on the wall changes style and location from when Bart writes it to when Homer walks by. (00:03:00 - 00:06:30)
Continuity mistake: At the start Marge's tiles go from on the table to in her hands when Homer chases Bart. (00:02:55)
Continuity mistake: When Homer chases Bart at the start, the book that Lisa had has disappeared from her hands and she is now holding her tiles. (00:02:50)
Continuity mistake: Milhouse makes a copy of Bart's photo for a brown skinned boy, but the boy's skin turns yellow when his father scolds him. (00:09:40)
Audio problem: When Lenny introduces Princess Kashmir, his mouth is open without moving when he says "presenting", and when he says "Princess Kashmir", his mouth only moves when he says "Princess", but is closed when he says "Kashmir." (00:07:50)
Continuity mistake: When Bart hits the streetlight while skateboarding, the streetlight is blue when he hits it, but turns grey as he lies on the floor. (00:08:50)
The Call of the Simpsons - S1-E7
Continuity mistake: When Homer is driving through the river, in one shot while Marge is talking, Homer's vest and shirt switch colors (from green shirt and grey vest to grey shirt and green vest). (00:07:15)
Continuity mistake: When Homer talks to Bart about ties, the two strands of hair on top of his head are missing. (00:09:20)
Continuity mistake: When Ms Melon is talking to Bart during the chemistry experiment, she is holding her clipboard with one hand on the back, but in the next shot she is holding the board with both hands on the side.
Continuity mistake: When Richard, Milhouse and Lewis are playing marbles in the school playground, behind Lewis there is only a tree, but in the next shot a table appears from nowhere.
Continuity mistake: When Homer goes to see Bart in his bedroom, the door goes from being open to closed between shots.
Continuity mistake: At lunch, two boys exchange Bart's sandwich for a grape. When they give him the grape, it lands in Bart's right hand but in the next shot he has it in his left hand, and when one boy gives Bart a cup, the grape is back in his right hand again, and in the shot after that, the two items have changed positions.
Continuity mistake: When Ms Melon is showing Bart his new classmates in the classroom at the Learning Center, you can see that Bart is standing in front of a wooden sculpture in one shot, but in the next shot he is standing in front of the bookcase.
Continuity mistake: When Homer and Bart arrive at The Enriched Learning Center, they go to the classroom and you can see there is a sign outside the room saying "Learning Center: Grade IV." But when we see the two a little later on, the writing has gone.
Continuity mistake: When Homer hands Principal Skinner the cheque, the back of the cheque is coloured yellow but changes to white a little later on. (00:07:20)
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