Continuity mistake: When Homer is wearing underwear while banging on the bathroom door the doorknob is missing, but next shot it appears on the door, and the closet door hinge also vanishes. (00:18:25)
Continuity mistake: When Otto's guitar playing wakes Marge and Homer they go to the garage. Note the direction of the door when Homer opens it toward the right, but when Marge asks Bart what's going on, the door's open the opposite way toward the left. (00:15:55)
Continuity mistake: When Marge comes downstairs she's shocked to see Homer in her house, and when the doorbell rings we see Mrs. Bouvier wearing an orange belt with a red buckle, but then the colors switch to an orange buckle and the belt center is red. (00:16:45)
Continuity mistake: When Homer stands outside the Bouvier house to pick up Marge, the home's front door has three diagonal windows from upper left to lower right. However, when Marge gets home later after the prom, the three windows are diagonal the opposite way from upper right to lower left. (00:16:10 - 00:21:00)
Continuity mistake: When Ned stops to snap at Lisa he's already yelled at Marge and Bart. Yet in the background you can see Marge among the secondary characters.
Continuity mistake: When explaining RDRR, Ms. Mellon writes "y=r^3/3" on the board behind her. But after she asks, "Don't you get it, Bart?", the equation disappears from the board and she writes it again.
Who Shot Mr. Burns? (2) - S7-E1
Factual error: Chief Wiggum visually compares a bullet taken from Homer's gun to the bullet removed from Burns and concludes that they are exactly the same. Excluding the obvious forensic violation, the bullet removed from Burns is still whole and within its casing, meaning that either Maggie took the time to remove the casing and delicately shove it in Mr Burns' bullet wound (which is dumb) or the gun fired the whole casing along with the bullet (which would have blown up the gun).
Mothers and Other Strangers - S33-E9
Continuity mistake: A teenage Homer and his father travel to Utah to look for Homer's fugitive mother, whom they find. This completely contradicts the episode Mother Simpson, where Homer didn't reconnect with his mom until he was in his late thirties, and was told all his life by his father that she was dead, even believing that the grave of Walt Whitman belonged to his mother.
Continuity mistake: In the scene at the United Nations, the Russian placard is just a placard. In the next shot, the placard is attached with a rod to rotate it at the push of a button.
Homer's Triple Bypass - S4-E11
Continuity mistake: When we first see Krusty start talking to Homer, from the rear shot you can see Krusty is missing his pale colour on his face. However, in every other shot, his pale complexion is there. (00:15:10)
Homer's Triple Bypass - S4-E11
Continuity mistake: When we see Lisa in Sunday school, the colour chair she is sitting on changes from blue to brown as she talks about her dad being very sick. (00:14:40)
Other mistake: When Moe is running to the Simpsons' house there is a shot of him running past the moon. When running back to the bar, the same shot is used but is simply flipped. Note the house and rocks. (00:18:20)
Continuity mistake: When Moe confronts Jimbo, Jimbo has his back in a corner with just a cabinet behind him. When the shot changes he is next to the green chair. (00:19:00)
Continuity mistake: When Bart is out in the hallway being attacked by the wolf, he is missing his right shoe. When he goes into class he is missing his left shoe. (00:18:40)
Continuity mistake: When we see Krusty with a bird on his head, the claws' position on Krusty's head change. Note his left eye. (00:16:20)
Continuity mistake: Smithers opens the second case for Tom Jones as the green gas has all but dissipated from the case. When the shot changes, there is still gas coming from the case. The remaining gas also changes place in the vicinity of the case. (00:15:50)
Continuity mistake: When Tom Jones walks off-stage a towel appears around his neck instantly between instant camera cuts. (00:16:00)
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