Continuity mistake: In the scene where Maggie swallow' Moe's penny, she first takes out her pacifier to eat it, but in the next shot it's in her mouth again and in the next, she's putting it back in her mouth.
Stop Or My Dog Will Shoot - S18-E20
Continuity mistake: In the scene where the dog chases Snake into a playground, Santa's little helper starts pulling the ropes off the top of the climbing frame, in one shot the dog has two ropes left but when we see Snake the dog chews through one rope and the whole thing falls.
Continuity mistake: While Bart is being beaten up by Nelson and Jimbo, Nelson is on the right side and Jimbo is on the left side. In the close up shot, you can see from their arms that Nelson and Jimbo have switched sides.
My Big Fat Geek Wedding - S15-E17
Continuity mistake: When Milhouse is hit by the rock in class, the map behind him changes colors. This actually happens within the same shot between frames.
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.
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).
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)
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)
Brother From Another Series - S8-E16
Continuity mistake: In the shot where Homer yells, "Marge, I'm going to look for the kids," their door is shut. After Homer sees the wave and runs back inside, their door is open all of a sudden.
Continuity mistake: When the robbery happens, Homer steps on the robber's feet, and we can tell he has large shoes. When they show video of the robbery on the news, the robber has regular shoes. During the hearing, they watch the video again. His shoes are back to long again.
Homer's Triple Bypass - S4-E11
Continuity mistake: When Krusty bursts into the room, the shot ends with him having his arms and legs bent. In the next shot, he's straight as a ruler. (00:15:10)
Continuity mistake: When Homer says he's going to the cinema, we see his drinking bird working in front of the computer. It makes a key press on the lower right of the keyboard, however on the following close up shot of the bird pressing the Y key it is a standard qwerty keyboard layout.
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