Continuity mistake: When we see the wide shot of all the houses going along the road, you can see that Moe's door is facing away from the Simpson's house. But in the next shot, Homer jumps in through Moe's door, it's like the tavern did a 180 degree turn.
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: 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.
Tree House of Horror VIII - S9-E4
Continuity mistake: Mutant Moe had one red mark on his face wile talking to Homer in the church, but when everyone attacks him, he has two.
Lisa's Sax - S9-E3
Continuity mistake: In the episode 'Lisa's Sax' (season 9, ep. 3), King Toot's music store is across the street from Moe's Tavern. However, in numerous later episodes, there are stills of Moe's Tavern that show King Toot's music store next door, not across the street as in the aforementioned episode.
Continuity mistake: When Ralph and Bart are playing cards, Ralph has four cards in his hand. In a following shot he has five then back to four cards again.
Continuity mistake: At one point Marge says "no guns at the dinner table" and Homer says "you said the breakfast table", then Marge replies "it's the same table". In most episodes they eat dinner at a different table.
Continuity mistake: When Ray Patterson's / Homer's office is shown, the text on the door is different according to the different shots. When the shot is on the door, the writing on the window (Name and Sanitation Commissioner) is close together, but when the shot is from inside the office with the door open, the writing is apart.
Continuity mistake: When "The Leader's" car drives by, trailing a cloud of dust, the dust doesn't appear on the people's robes until the cloud clears away.
Bart Carny - S9-E12
Continuity mistake: When Principal Skinner is playing the ring toss game, in the end he only has one ring left in his hand. Then, after his mother asks "Why do you always fail?", he throws two more rings trying to win the combat knife.
Continuity mistake: As soon as the family has left the Knowledgeium, they are immediately outside walking away. The shot changes and the building is nowhere to be seen.
Continuity mistake: Bart turns the door handle to the joke shop and a boxing glove comes out through the door and hits him. But when he opens the door there is nothing on the other side for the glove to come out of.
Bart Carny - S9-E12
Continuity mistake: Up until the point at which Skinner starts throwing rings at the block in front of the Spanish lamp, the block simply doesn't exist. It only appears once he starts throwing the rings.
Continuity mistake: When Chief Wiggum finally climbs the stairs his gun is visible to be a black, old style revolver with a curved handle. In the next shot where he draws his gun, it is a grey, new style handgun.
Continuity mistake: When Homer is running down the street with his garbage, it spills out all over the road. But in the next shot when the garbage men are confronting him, all the garbage is gone.
The Two Mrs. Nahasapeemapetilons - S9-E7
Continuity mistake: The amount of liquified chips in the bag at the home changes depending on the shot.
Continuity mistake: When the family are on the conveyer belt at the beginning the family are travelling along. In the next shot (the close up) they are travelling alot slower.
Bart Carny - S9-E12
Continuity mistake: When the Simpsons drive up to the house, there are no boards on the window. When they can't get in, suddenly there are boards on the windows.
Realty Bites - S9-E9
Continuity mistake: While Homer is driving Snake's car, he passes a place called Heinrich's Monocle Shop three times.
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