Homer's Barbershop Quartet - S5-E1
Other mistake: When the Be Sharps are singing at the Statue of Liberty, in some shots the stars in the sky are black.
E-I-E-I-(Annoyed Grunt) - S11-E5
Revealing mistake: After Lisa says "For a product that's evil and deadly" while standing in front of the tomacco stand, there's a duplicate Lisa behind her that stands still while she moves, giving Lisa the appearance of having two heads. It looks like the animators made a mistake while trying to animate Lisa.
Factual error: In the scene where Lisa's great great grandmother, and the slave Virgil escape to Canada, they enter Canada across a suspension bridge, which has, on the Canadian side, the Canadian Maple Leaf flag. In 1860, Canada was not yet even a country, and the flag was the Union Jack and the Canadian Ensign. The Maple Leaf was not adopted as the Canadian flag until February 15, 1965.
Brother From Another Series - S8-E16
Continuity mistake: In the scene where Sideshow Bob and his brother Cecil are arrested and sits in the the back of the police car, Cecil is missing when Sideshow Bob is talking to the people outside the car though his window. They both entered the car and they can both be seen when the car leaves.
The Secret War of Lisa Simpson - S8-E25
Continuity mistake: When Lisa has just been saved from falling, from the Eliminator at night, three boys approach. The two boys on the right swap positions between shots.
Continuity mistake: The hole in the wall made by the acid changes shape and size in several scenes.
Lisa on Ice - S6-E8
Continuity mistake: The turtle that Homer hands to Bart disappears as soon as it goes into Barts' hand, and it is never seen again.
Continuity mistake: When Millhouse spits the paper at Kearney in the cafeteria, Jimbo is seen wearing a black shirt and Dolph is wearing a green shirt yet, when Millhouse is dragged out the cafeteria by Bart, they are both wearing blue shirts. (00:16:10)
Continuity mistake: In the shower scene, we see that in front of Carl, there is a man with brown hair and a moustache in front of him, but then when the shot changes, the man has moved cubicles and a man with white hair has taken his place, and he wasn't even in the previous shot. (00:14:50)
Continuity mistake: When Chief Wiggum says, "Give him your badge," he doesn't have anything on the right side (his right) of his shirt. After the camera flashes to Bart and back, Wiggum suddenly has a badge there.
So It Has Come to This: The Simpsons Clip Show - S4-E18
Audio problem: When Lisa explains the origin of April Fools Day, we hear Homer, Marge, Bart and Lisa chanting "Blood For Baal." But, we see that Lisa is not taking part in the ritual and her mouth isn't moving. (00:02:15)
Continuity mistake: When Marge talks to cable guy, the trees and hammock disappear.
Continuity mistake: When the family are returning from the picnic, they pull into the drive and Homer has a seatbelt on in some shots, and in others he isn't wearing it. You can't even see it attached to the side of the car sometimes.
Continuity mistake: In class, when Lisa is sitting at her desk, pondering the Jebadiah mystery, her tray of cookies and glass of milk is very visible. Then when Lisa says "The missing piece of the puzzle. I've got it." and runs out of class, the cookies and milk are no longer there.
Milhouse Doesn't Live Here Anymore - S15-E12
Continuity mistake: Right at the beginning of the episode when all the children are on the bus we can see that in front of Nelson is Bart and behind him are Lewis, Martin and Wendall. Then when the shot changes to behind Nelson, Lewis has swapped places with Bart, Bart has disappeared and some weird kid is sitting where Lewis was. (00:00:40)
Continuity mistake: When Homer first finds the corpse, there is no belt on the trousers it is wearing. But in the shot where Homer starts to scream, there is a belt on the trousers.
Continuity mistake: When the plane takes off without them, you see that the Simpsons are only carrying 2 bags and 2 suitcases. But when it cuts to them un-packing in the house, there are 5 suitcases.
Continuity mistake: Willie rips off his entire shirt before he fights the wolf, but in the next scene, when he is wrestling the wolf, he is wearing a torn-up shirt. (00:18:40)
Who Shot Mr. Burns? (1) - S6-E25
Continuity mistake: When Burns is pretending to be a student in Skinner's office, the door changes from open to closed and from dark blue/green to grey.
Other mistake: During the shot when Marge asks Homer who 'Hank Jones' is, you can see that the couch curves down on the left side of the screen, therefore putting her on the other side of the couch. (00:07:40)
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