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: When lying on the sofa at the start, Bart's raised leg switches between instant cuts. (00:02:30)
Revealing mistake: When Bart is swinging on the clothes line into the night, it is obvious that the brightness has been turned down because the sky is the same colour as in the previous shot. If it were night time, then the sky would be a dark purple colour. (00:04:20)
Continuity mistake: In one scene Bart is swinging on a washing line then we see Homer talk to Marge and he says "He'll soon get tired of it" then it goes back to Bart and he is facing the opposite direction and he is swinging on two washing lines. (00:04:22)
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)
Revealing mistake: Part of Homer's right arm flashes white when he asks his father for $15,000. (00:08:50)
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: After Krusty is done doing the special promotion commercial, he is given a telegram informing him of the Soviet boycott. However, this is a flashback, and at this point in time Krusty is illiterate (as will later be established in season 1) so he shouldn't be able to read the telegram. (00:12:30)
Continuity mistake: Rodd's hair is black when him and Todd are in bed singing. It can't be because it's dark, because Todd's hair is unchanged. (00:15:10)
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.
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