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)
Continuity mistake: In the title sequence Mr. Largo has a baton in his hand when conducting. This disappears as he stops conducting to throw Lisa out. (00:00:50)
Continuity mistake: At the end, we see Krusty stood next to Mr. Burns. However, as the camera pans over the crowd he has vanished. (00:20:30)
Continuity mistake: Lisa's posture changes at the breakfast table as Bart talks about the school bus. (00:16:10)
Continuity mistake: During the theater scene, the man in the back with the moustache and green jacket watching the Space Mutants movie changes from having yellow skin to brown skin after Bart blows a raspberry. (00:09:30)
Continuity mistake: The position of the street lamp Bart skates into changes between shots. Notice the lines in the pavement. (00:08:20)
Continuity mistake: When the car pulls up by the theatre, you can see it through Bart's window. As the car drives off it's now trees outside his window. (00:08:00)
Continuity mistake: When Marge is telling Homer off about staying in the car she has gloves on. Camera changes to her saying "Move it" and now her glove has disappeared. Her hand also moves from being on the door to pointing. (00:05:00)
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)
There's No Disgrace Like Home - S1-E4
Continuity mistake: When Homer walks in on the family eating dinner, Marge has her feet on the second cushion of the sofa. Camera changes and now they are on the third cushion. (00:08:50)
There's No Disgrace Like Home - S1-E4
Continuity mistake: When they put Maggie down in the Creche she is in the middle of a carpet. Camera cuts and now she's at the edge. (00:04:25)
There's No Disgrace Like Home - S1-E4
Continuity mistake: The amount of marshmallows in the jelly at the start changes between shots. (00:02:00)
Continuity mistake: When Sherri and Terri raise their hands and say "like us Mrs. Krabbapple" they raise their hands over their heads. Camera cuts and their hands are a lot lower. (00:04:20)
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)
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 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)
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: When Bart is in Principal Skinner's office, there is a voice box on the right hand side of the desk. But when Principal Skinner looks into the filing cabinet, the box has gone. (00:06:40)
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)
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