Continuity mistake: In the bit where they sit on the sofa at the start of each episode, there is one rather long one where it starts off with the Simpsons dancing in a line, then gradually evolves into a circus with elephants and men dancing around fire. Look closely in between Homer's right leg and the dancing woman on his right. There is a pile of books on the little shelf underneath the lamp and telephone, and just as the camera starts zooming out from the line of dancing, the orange book on the left noticeably changes to purple. (00:01:20)
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?
Homer's Triple Bypass - S4-E11
Other mistake: In the beginning of the episode when Chief Wiggum orders a battering ram through The Front door of a house, Reverend Lovejoy opens the door and questions what they are doing. Chief Wiggum says "isn't this 742 Evergreen Terrace?" to which Lovejoy replies that it's the house next door. In the next shot, you see Snake pulling out of the garage and speeding away. Everybody knows that 742 Evergreen Terrace is, in fact, the Simpson residence. (00:02:40)
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.
Audio problem: Just after Homer says "About 3 Weeks" in reply to Bart's question about how long Grandpa lived with them, the family all start laughing. However, the second time Homer laughs, his mouth doesn't move.
Homer's Triple Bypass - S4-E11
Continuity mistake: Homer is interrupted from praying by a nurse who comes into his room and says, "Ssh." However, when she says this, the door is closed, and it is only in the next shot where the door is open immediately and she is pointing to the NO PRAYING sign. (00:14:20)
Continuity mistake: When Marge and Homer are arguing about Don Quixote, Marge's pillow disappears from under her head. (00:09:50)
Tree House of Horror III: The Simpson's Halloween Special III - S4-E5
Audio problem: When Lisa is turned into a snail in the Bart's zombie story, she says "awww" but her mouth doesn't open. (00:20:25)
Tree House of Horror III: The Simpson's Halloween Special III - S4-E5
Deliberate mistake: In "Clown Without Pity", when the Krusty doll accidentally knocks the head off of Lisa's Malibu Stacy (her doll), Krusty puts it back on but you don't see a crack where the head came off. (00:07:40)
Continuity mistake: While Homer is sleeping, the fire burns off the hair on his head. But when he wakes up and jumps up off the couch, his hair is back again. (00:17:00)
Continuity mistake: In the episode "New Kid on the Block," Bart says his home address is 1094 Evergreen Terrace. In "Homer the Vigilante," his home address is 723 Evergreen Terrace; and for the rest of the series, it has been 742 Evergreen Terrace. (00:18:30)
Kamp Krusty - S4-E1
Factual error: In the last scene, with the picture of the bus leaving Tijuana, it says "Leaving Tijuana" above the gate leading into the city.
Itchy and Scratchy: The Movie - S4-E6
Audio problem: On the Itchy and Scratchy Movie ad, Itchy rips Scratchy's brain apart. He is laughing as well but his mouth doesn't move.
Last Exit to Springfield - S4-E17
Continuity mistake: When we see McBain at the beginning of the episode, after shooting some people, the magazine falls out of the machine gun. But from the next shot onwards it is back in the gun.
Continuity mistake: When the flashback of Ned and Todd Flanders singing in the Simpsons' doorway, Homer says he could use a TV tray. Then it goes back to the present time and Homer picks up a beer from the TV tray which is next to the couch. The TV tray was never there at any other point in the show. (00:10:40)
Continuity mistake: When Krusty the Clown arrives at the beauty pageant his tie is done up. When he goes on stage it is undone.
Tree House of Horror III: The Simpson's Halloween Special III - S4-E5
Continuity mistake: When Homer is escorting his family out of the house to escape the Zombies, he leaves the door wide open, but in the very next shot it is shut. (00:18:50)
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