Continuity mistake: When Roger Meyers throws his nameplate at the writer the second time, the animation is repeated from when he threw it at him earlier. It's more obvious because his mouth doesn't match what he's saying and the background is orange instead of purple like the rest of the scene.
Continuity mistake: Millhouse is in the classroom at the beginning when Principal Skinner says that superintendent Chalmers is coming for a surprise inspection, but when they show Nelson he is getting crud put down his shirt. How could he be at two places at once?
Continuity mistake: On 'Whacking Day,' Homer is seen charging people to park on his lawn. But later when the snakes are lured to the Simpson house, there are no cars there.
Continuity mistake: When the bottle of bourbon is found and taken out of Marge's pocket, the bottle is grey. But when Sanjay throws it to Apu, and Apu catches it, the bottle has now changed to a brown colour. (00:08:30)
Continuity mistake: When Marge comes to pay for her shopping, she empties her left pocket on her coat, but when Sanjay finds it in her pocket, it's in her left pocket. We know it wasn't in her pocket anyway because when Sanjay finds it, it is bigger than the size of her pocket.
Audio problem: At the end of the episode the statue of Jimmy Carter with Marge's hairdo is unveiled, and the assembled throng applaud her reaction. They are all chattering away, but no sound is coming out of their mouths - we only hear the sound of clapping.
Krusty Gets Kancelled - S4-E22
Other mistake: When the Red Hot Chili Peppers are performing on Krusty's TV special, bassist Flea is playing a standard electric guitar.
Krusty Gets Kancelled - S4-E22
Continuity mistake: When Bart and Lisa are helping Bette Midler pick up trash on the highway, Bette picks up an empty can which says Buzz Cola on it. Just as she yells, "You pigs!" the label on the can switches to Duff Beer.
Homer's Barbershop Quartet - S5-E1
Character mistake: When Marge is making a model of Homer for Bart and Lisa, she makes him bald with his hair round the back of his head as we all know it, but at this point in the story Homer was not yet bald. (00:18:15)
Homer's Barbershop Quartet - S5-E1
Continuity mistake: When you see a shot of the church from outside in the scene where the bee sharps sing there, look closely at the sign. It says 'Also: The Bee Sharps', only they don't name themselves that until later on in that episode.
Homer's Barbershop Quartet - S5-E1
Continuity mistake: In the flashback where Marge is making a model of Homer, the models head falls off and Santa's Little Helper picks it up, but the Simpsons did not own the dog back then.
Homer's Barbershop Quartet - S5-E1
Audio problem: When the Bee Sharps are singing 'Goodbye my Coney Island Baby' next to the Statue of liberty, you can hear Chief Wiggum's voice singing with the other members of the group, but he got chucked out of the group earlier on in the episode.
Homer's Barbershop Quartet - S5-E1
Continuity mistake: At the sale, when Bart and Lisa first find 'The Bee Sharps' album, the order of the records in the stack change in subsequent shots.
Homer's Barbershop Quartet - S5-E1
Audio problem: When Homer is playing on the piano, he says the word "fault". When he says this you can hear a key being pressed but Homer doesn't touch any keys.
Homer's Barbershop Quartet - S5-E1
Continuity mistake: At the end of the episode, while the B-sharps are singing on top of Moe's Cavern, the guy climbing the building asks the people to look at him. As he moves his hands, his gloves change color from red to white and back to red.
Homer's Barbershop Quartet - S5-E1
Audio problem: When grandpa is listening to the radio, he says something like 'oh.', but his lips are not moving.
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.
Homer's Barbershop Quartet - S5-E1
Audio problem: After Jasper has sung on the stage to Homer, Apu and Skinner, they simultaneously say "Next". But if you listen carefully you can't hear Homer's voice.
Homer's Barbershop Quartet - S5-E1
Other mistake: When Bart and Lisa are looking at records, there is a picture of Homer. He had hair at the time the record was made, and the hat he's wearing isn't big enough to cover it, but there's none visible as there should be.
Homer's Barbershop Quartet - S5-E1
Continuity mistake: When Homer is talking to George Harrison, he sees that he has a brownie. Then he rushes over to a pile of brownies and starts scoffing them down. The pile of brownies doesn't get smaller.
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