Much Apu About Nothing - S7-E23
Continuity mistake: When Bart pulls out Apu's punch cards, there are many sitting on top of the box. When Apu picks the box up and places it in the bin, there are only a few cards sitting on top.
The Ziff Who Came to Dinner - S15-E14
Continuity mistake: In once scene, Artie walks past the Simpson's to Selma (or Patty). In the next shot he is shown walking past the Simpson's again.
Alone Again Natura-Diddly - S11-E14
Continuity mistake: In the episode where Maude dies, Homer puts Ned to bed in the right hand side. When Ned tells Rachel he's ready to smooth out Maude's shape, they smooth out the right hand side.
Continuity mistake: In the scene where Bart and Homer find Marge's paintings in the attic, there is a shot of the paintings all stacked up and you can see that the edges of all of them are the same shade of pink as The Front painting. However, when the boys flick through the paintings, the backgrounds and edges are a variety of colours, including pale blue.
Like Father, Like Klown - S3-E6
Continuity mistake: During Krusty's performance, the tables change from round to long and rectangular. (00:09:10 - 00:10:10)
Continuity mistake: At the end of the episode, when we see a wide shot of everyone going to join Ned on the hill (and singing the song) look to the far left. You will see Apu - and his skin is yellow.
There's No Disgrace Like Home - S1-E4
Continuity mistake: When the Simpsons are spying on the old man, the boy and the old woman, all three of them disappear and then reappear between shots. Whenever there is a wide shot from outside the house, they are not there but when there is a close shot from outside or inside they are there. (00:11:00)
And Maggie Makes Three - S6-E13
Continuity mistake: When Homer is trying to get Ned another woman in a different episode, he tells him that there was some footage of Maggie being born that he couldn't get rid of on the tape but in the scene (in this episode)when Maggie is being born there are no video cameras in sight.
Continuity mistake: When the family is at dinner, there is a top down view which zooms in on them from above. In this shot there is no newspaper on the chest (to Homer's left). In the next shot of Homer a newspaper has appeared. (00:07:50)
A Streetcar Named Marge - S4-E2
Continuity mistake: As Marge is about to leave for her first audition, she walks across the room and she is wearing white slip-on shoes. Then her and Homer talk for a bit, then when it cuts to a wider shot after they talk, she now has her red flat-heels on. (00:03:30 - 00:04:00)
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.
Continuity mistake: When Grandpa goes nuts outside the house he knocks over 2 bins and spills rubbish on the pavement, in the next shot the bins and the rubbish are gone.
Continuity mistake: When Lisa meets the other kids for the first time under the boardwalk, her tinted shades disappear in the very last shot.
Lisa's Pony - S3-E8
Continuity mistake: When Lisa calls Homer at work, you can see some monitors to his right, and a fire hydrant in back of him. When he turns to his right, you can see that the monitors disappear, the fire hydrant moves, and a green telephone appears next to it.
Continuity mistake: When Homer is writing a note to Marge, he scribbles out Patty's name and writes his own name directly beneath it. When Marge reads it, Homer's name is under Patty's, but slightly to the right. (00:14:30)
Team Homer - S7-E12
Continuity mistake: During the classroom scene where Bart reveals his Mad magazine shirt, he removes his sweater and puts it on his desk. In the next shot the sweater disappears. (00:03:40)
Sweets and Sour Marge - S13-E8
Continuity mistake: When Lou throws a small chocolate statue in the fire, his skin is yellow when it is usually brown.
Homer's Triple Bypass - S4-E11
Continuity mistake: When Dr Nick visits Homer for the first time, he leaves the door open. But later before the reporters come to the door, the door is closed. (00:13:40)
Brake My Wife, Please - S14-E20
Continuity mistake: Whilst Moe is talking to himself about going back to barber college the door to his bar is wide open. When the shot changes to him when he says "Who the hell am I talking to?" the door has suddenly closed. There was not enough time for it to close between shots.
Home Away From Homer - S16-E20
Continuity mistake: For "Home Away From Homer": Bart looks out his bedroom window and sees the college girls doing sexual actions in the Flanders' spare bedroom. But, there is never a window in Bart's room that could possibly face the Flanders' house. [Bart's window in many episodes does face Flander's house but in other episodes like "Bart's Friend Falls in Love" and "Bart the Mother" it faces the treehouse which is the backyard of the house.]
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