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.
Revealing mistake: When Moe grabs Ned Flanders' shirt at the bar, Ned's torso is incorrectly drawn over his arms as well as the mug he is holding. (00:10:50)
Simpsoncalifragilisticexpiala(Annoyed Grunt)cious - S8-E13
Continuity mistake: When Bart is lying on his bed whipping cupcakes at the wall, there are already 2 on the wall but after he whips the 3rd, there are still only 2 cupcakes on the wall.
Grandpa vs. Sexual Inadequacy - S6-E10
Continuity mistake: When the janitor shows Marge and Homer into the Utility Room, the only equipment in there is a couple of mops. Later, the janitor comes back and pulls out a big vacuum cleaner that we never saw previously.
Grandpa vs. Sexual Inadequacy - S6-E10
Continuity mistake: At the retirement home, Grandpa pulls out some medicine from his cupboard. In the next shot, another medicine bottle that wasn't in the previous shots appears.
Grandpa vs. Sexual Inadequacy - S6-E10
Other mistake: When Grandpa throws the bottle to Homer, look at the bottle - at the angle it was thrown at, it should have missed Homer completely.
Continuity mistake: In the full view of Mr Burns' office when he is talking to Homer towards the end of the episode, there is only one fountain pen holder on the desk, but when Mr Burns tells Homer to leave after giving him his old job back, there are now two holders on the desk.
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