Continuity mistake: When Bart is writing from the books in the kitchen, he is seen writing on the paper. In the next shot of him, the paper is clean.
Continuity mistake: In the opening shot of Apu serving his customers, you can see Martin wearing a purple shirt. However, later when all the customers are grumbling about the price, you can see Martin walk away wearing a white shirt.
Continuity mistake: Apu grabs the ham and crosses out of the expiration date with a thick black marker. When he places the ham on the shelf the black marker cross out is gone and the writing on the pack looks perfectly printed as before.
Continuity mistake: When the robber shoots Apu, he is shot around his leg. In the next shot, the wound is in his chest.
Lisa vs. Malibu Stacy - S5-E14
Continuity mistake: During both conversations Lisa has with the real Stacy (the creator of Malibu Stacy), her drink magically refills itself a number of times.
Lisa vs. Malibu Stacy - S5-E14
Continuity mistake: When Lisa and Marge go on the tour, when the film has finished the woman turns it off with a remote controller and puts it behind her back. In the next shot from behind her it isn't in her hands and there is no sign of a pocket or any indication she put it anywhere else.
Lisa vs. Malibu Stacy - S5-E14
Continuity mistake: When the Simpsons are leaving the house to go to the mall. The aerial shot shows that they are leaving in Marge's red station wagon, but the very next shot of them in the car shows that they are in Homer's pink sedan. The shot of the station wagon leaving is probably a re-used shot from a previous episode.
Lisa vs. Malibu Stacy - S5-E14
Continuity mistake: When the Simpsons are about to leave the house to go to the mall, they're all wearing jackets, but when they're in the car just a few seconds later, none of them have their jackets on.
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: When Barney becomes his old self again at the training center, he gets on a jetpack while holding a bottle. In the next shot while he is in the air, the bottle is gone.
Continuity mistake: At the end of the episode in church when the group of parishioners moves forward to forgive Ned there is a woman amongst the group who has light green hair. However, when there is a wide angle shot of the group again she is nowhere to be seen.
Continuity mistake: When Homer drives his car to work it is lime green instead of the usual purplish color.
Continuity mistake: When Ned is saying grace, the plates on the table contain no food, but when Homer comes through the window there's food on the plates.
Bart Gets an Elephant - S5-E17
Continuity mistake: Immediately after being chained to a tree, Stampy crushes several objects (a lawnmower, a bike, a doghouse, and a trash can). In the very next shot, which comes after a commercial break, all of these objects are in fine shape. Then the shot changes again and we see them destroyed once again. (00:09:30)
Bart Gets an Elephant - S5-E17
Continuity mistake: Bart's elephant crushes the dog house, then it is shown standing, and then it is crushed again.
Burns' Heir - S5-E18
Continuity mistake: When Mr. Burns is sitting in front of the fireplace, he has his normal blue outfit when seen from behind, and a purple night gown when seen from The Front. (00:02:50)
Burns' Heir - S5-E18
Continuity mistake: When Homer and Marge are talking to Chief Wiggum, the Rubik's Cube and the cup on his desk changes to a doughnut between shots. (00:13:20)
Burns' Heir - S5-E18
Continuity mistake: When Homer says, "You idiot," to the programmer, the red ashtray changes position on the desk. (00:15:55)
Burns' Heir - S5-E18
Continuity mistake: The doughnut Homer later picks up isn't by the door when Lenny walks into Mr Burns' office, and we never see anyone put it there.
Burns' Heir - S5-E18
Continuity mistake: Hans Moleman has brown skin at the beginning of the episode and yellow skin at the end.
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