Continuity mistake: In the cafeteria scene, the shots from Lisa's perspective show glass protectors with signs where the food is. However, when the shot switches to Lunch lady Doris' perspective, the protectors are gone.
Treehouse of Horror XVIII - S19-E5
Continuity mistake: When Marge shoots Chief Wiggum, the arrow goes through his tie. When they roll on top of him, the arrow isn't through the tie anymore.
Continuity mistake: The door to Mrs. Krabappel's classroom has a window, but when Bart fakes being sick then walks out of the class (we see the desks as the door closes), the door's window has vanished, and the doorknob has switched sides as well. Additionally, earlier, when Bart deliberately falls off his chair in class, note that one of the bars on the back of his chair has gone missing. (00:07:35)
Revealing mistake: When Marge stands up at one point in the airplane restaurant, her legs and part of her dress have disappeared.
Please Homer, Don't Hammer 'Em - S18-E3
Continuity mistake: As the workmen begin to leave site after hearing that Homer is unable to pay them, a couple of tools are dropped on the ground; however, in the following shot, facing back towards the cabinet where Homer is hiding, the tools have disappeared.
Continuity mistake: After Mrs. Krabappel asks Bart to name the pirate in Treasure Island, when it cuts to a closeup of Bart we see the set of blue books have vanished from Mrs. Krabappel's desk, and a Krusty mug has suddenly appeared there. (00:03:00)
Continuity mistake: When Bart fails the test despite having actually studied, we see the two red "X" marks in the center row of the multiple choice questions, but after Mrs. Krabappel passes Bart because he applied practical knowledge, when Homer hangs the test up on the fridge those two "X" marks in the center have vanished.
Continuity mistake: Bart's bedroom is on the opposite side of the stairs, with his window facing out to the backyard, but after his prayer his bedroom window is facing The Front of the house. Additionally, the style of his window changes as well, going from two horizontal sliding panes with no muntins, to a vertical sliding window with two muntins when Marge pulls the curtain back, then to a having only one muntin when Bart opens his window.
Continuity mistake: When Bart is called up to give his oral book report keep your eye on Bart's book, Treasure Island, and the direction of its binding. The book's binding should be to the left of The Front cover, which has the picture of a pirate wearing a black hat, but the binding changes to the right side (note the cover picture/title) of the cover at times, and when he mentions the publisher the book also changes how it's held by Bart. (00:02:10)
Grandpa vs. Sexual Inadequacy - S6-E10
Continuity mistake: On the book on conspiracy theories that Bart buys, there's a picture on the back of the author. The picture changes several times throughout the episode.
Grandpa vs. Sexual Inadequacy - S6-E10
Continuity mistake: When Homer and Grandpa are trying to sell the tonic at the mall, when Homer is trying to sell tonic to the Critical Man, all the tonic bottles on the table are actually empty. In the next shot, they're full.
Grandpa vs. Sexual Inadequacy - S6-E10
Revealing mistake: When Doctor Hibbert drinks the bottle of tonic, the amount of tonic never actually changes even though it should have gone down a considerable bit.
Grandpa vs. Sexual Inadequacy - S6-E10
Continuity mistake: When Lisa jumps off the tyre swing, her dress is fairly dirty. When Homer drives off, the bottom is dirtier than before.
Grandpa vs. Sexual Inadequacy - S6-E10
Continuity mistake: At the old Simpson home, Grandpa and Homer run into each other and collide on the porch. In this shot, the windows are dark, but in the next, all the windows are instantly lit up by the fire, which would not be possible in the given time.
Continuity mistake: When the snake from the cashier bites Moe, the bites on his arm never change and only remain in the one spot even though the snake bit several different spots on his arm.
Continuity mistake: In the scene in the airport bar, two pilots sitting next to each other have brown and gray hair, respectively. When they stand up, the hair has switched colours.
Continuity mistake: On the plane, where Marge is starting to act up due to her Fear of Flying, Homer and Marge's seat-belts have mysteriously disappeared.
Continuity mistake: When we first see Homer at Moe's at the beginning of the episode, he is sitting at the bar and there is an ashtray by his glass of beer. But after a quick cut to Moe, we return to Homer and the ashtray has disappeared.
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