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.
The Last Temptation of Homer - S5-E9
Continuity mistake: Mr Smithers throws a notepad at the person saying that a real emergency exit is required, but when we see it on the floor it is now a book with a green cover.
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.
Secrets of a Successful Marriage - S5-E22
Continuity mistake: When Homer gets thrown out of the house by Marge and Otto walks up to him, you can see that it is Smithers and Carl standing behind Homer. Yet in the next shot when you see Homer's face, it is now Principal Skinner and Carl standing there.
Burns' Heir - S5-E18
Continuity mistake: When Homer and Marge are talking to Chief Wiggum, the Rubik's Cube and the cup on his desk changes to a doughnut between shots. (00:13:20)
Continuity mistake: The man conducting the tour of the box factory tells the children that they will not be able to see a finished box, as the boxes are assembled in a separate location. However, moments earlier we saw a flat box going through a machine, and coming out finished.
Continuity mistake: When the person delivers the animation cell, his skin changes from white, to brown and then back to white. When Bart looks through the letterbox to see who was there, his skin is white. When he gives the box to Bart, it is brown. When he hits Bart in the face and says "that's for keeping me waiting", it is white again.
The Last Temptation of Homer - S5-E9
Continuity mistake: When Homer and Barney are at Moe's, Barney drinks his beer and then his cup his empty, and Homer's is full. Then later, Homer's cup is empty and Barney's cup has magically refilled itself.
Continuity mistake: Just after Homer has used the proton accelerator, Dean talks to him. In this scene he is writing something on a piece of paper, but he writes on the same line every time. He finishes writing on that line, then starts on it again.
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.
The Boy Who Knew Too Much - S5-E20
Continuity mistake: Hans Moleman's clothes change from white to black during the trial. (00:12:05 - 00:13:25)
Burns' Heir - S5-E18
Continuity mistake: The doughnut Homer later picks up isn't by the door when Lenny walks into Mr Burns' office, and we never see anyone put it there.
Continuity mistake: When we first see the power sander, there is only a white patch on the side. In the next shot when we return to viewing it, "Property of Ned Flanders" has appeared.
$pringfield (or How I Learned to Stop Worrying and Love Legalized Gambling) - S5-E10
Continuity mistake: When Marge comes home after the bogeyman incident, Homer trips over a footrest. You can see Bart sitting there however his pyjama's are a dark green color when before they were a bright green color. (00:16:35)
Continuity mistake: When Homer is shouting at Skinner about losing Bart, if you look closely you can see that the box with the red cap goes past Homer. However, when he turns round and sees that red cap on a box, that box has already gone by.
The Last Temptation of Homer - S5-E9
Continuity mistake: When Bart has had his throat sprayed, we see him standing up just before the shot changes to the power plant. The shine on his shoes instantly disappears and reappears several times.
Continuity mistake: When Marge and Ruth and shooting cans, there are five on the fence, then, in the next shot, there are only four.
Secrets of a Successful Marriage - S5-E22
Continuity mistake: When Marge is shopping at the Kwik-E-Mart she puts a red can and a blue can on the counter.When Apu tells her about the offer on hair dye he puts two blue cans into the paper bag.
Continuity mistake: When Homer and Marge are on stage with Brad Goodman, Homer is eating toffee apples. He keeps eating them but they stay the same size.
Lisa vs. Malibu Stacy - S5-E14
Continuity mistake: During both conversations Lisa has with the real Stacy (the creator of Malibu Stacy), her drink magically refills itself a number of times.
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