The Trouble With Trillions - S9-E20
Continuity mistake: In the scene where the FBI agent and Homer are going into the photo booth to view the video, look to the right of the photo booth before they enter. There is a white cup next to the booth. After they have watched the video and are standing outside the booth talking, the cup is now red instead of white.
Continuity mistake: When Bart and Ralph twist the key into the armed switch, the armed sticker is red and the off sticker is yellow. Then, during the press conference, the stickers switch colors. Then, when Mayor Quimby sits in the electric chair, they switch back.
Last Exit to Springfield - S4-E17
Continuity mistake: When the dentist shows Ralph the Big Book of British Smiles, he first shows him a picture of Sherlock Holmes with horrid teeth and all of his teeth are there. When Ralph starts getting terrified of it, it shows the same picture, but this time one of his teeth on the bottom right hand corner of his mouth is missing.
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)
Continuity mistake: When Agnes meets Bart and his friend, it cuts to Skinner, then to Bart's group and quickly back to Skinner. When it goes to Bart's group (before Skinner) Lewis is yellow like the rest, in all other scenes he is black.
Continuity mistake: Just before Lisa's Presidential "refund adjustment" speech she pushes Bart's tape player off the desk and we can hear it smash off screen. Later in the episode when Lisa agrees to legalise it, he starts playing the same player.
Continuity mistake: During The Town hall scene, Ruth Powers (the woman with the red bandana) is seen with blondish hair in two shots. All other shots of her during this scene shows her with her usual brown hair.
Itchy and Scratchy: The Movie - S4-E6
Revealing mistake: When Nelson blinks right before Milhouse says "No one who saw the movie would say that", Nelson's right eyelid is white for some reason. (00:18:20)
Tree House of Horror II: A Simpsons Halloween - S3-E7
Audio problem: When the guy who Homer buys the monkey's paw from tells him that he'll be sorry, his mouth doesn't move. (00:03:10)
Continuity mistake: When the women of the neighbourhood throw Marge a baby shower because she's pregnant with Maggie, Ruth Powers is one of the ladies there. Ruth only met Marge when she moved to Springfield with her daughter in "New Kid on the Block" (Maggie was already born in the episode).
Homer's Triple Bypass - S4-E11
Factual error: I get it, it's a comedy. But after open heart surgery, you are sedated for 1-2 days and on a ventilator. So, Homer even being awake after open heart surgery is completely false. Also, as someone who has had open heart surgery, you can barely move your chest because your sternum is fractured.
The Last Of The Red Hat Mamas - S17-E7
Factual error: Milhouse says he speaks fluent Italian, but his Italian is full of basic grammatical errors. He says that "insegnante" means "masculine teacher" when the same word is used for a male or female teacher. There are errors in Lisa's Italian course too ("barca piccola" instead of "piccola barca"). Despite Lisa's understandable mistakes, as she is a learner, Milhouse never corrects her (there is no such word as "idioto"). Even Milhouse's grandmother, who is Italian, makes many grammatical mistakes.
Other mistake: At the finish line, the crowd are missing all their appearance features such as faces, hair etc. (00:13:50)
The Day the Violence Died - S7-E18
Other mistake: At the end of the episode Lisa and Bart watch Itchy and Scratchy in the TV room, which is beside the kitchen at the back of their house. The TV room has a window that faces the backyard, with the record player on a shelf in front of that window. The problem is when Bart looks out the back window and sees Lester skateboarding past the house, the view outside is actually the street in front of the Simpson house, not their backyard as it should be.
Other mistake: At the concert when Bart throws the Frisbee into the dude sat next to him, the crowd in the background are all missing their eyes. (00:03:50)
Continuity mistake: When Marge spots Krusty, from the regular view, Krusty's hand is down on the table. But from the view from the camcorder, his hand is on top of the cake. (00:14:00)
Other mistake: In the scene set in the future after Lisa, as the US President, held a conference with some journalists, just before the scene changes they show a black journalist with eyes not drawn among the crowd.
Continuity mistake: When we see Marge leave Bart with Patty and Selma, they are stood with a gap between them. Camera changes to focus on the sisters, and they are now next to each other. (00:06:30)
Other mistake: When Homer pretends to read the newspaper, the back page is very noticeably blank and missing any print. (00:08:00)
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