Tree House of Horror II: A Simpsons Halloween - S3-E7
Continuity mistake: When Mr. Burns is telling the robot Homer that he is his daddy, Burns is to the right of Homer, but after the POV of Burns' face (he is still oriented right of Homer) by Homer, Burns is now to the left of Homer.
Continuity mistake: When Homer ignores Lily, she gets upset then runs out of Moe's, and in the interior shot the door is hinged at the left, but in the next exterior shot that door is hinged on the opposite side, while Lily is crying outside. (00:09:35)
Continuity mistake: When Marge walks out of her aunt's house onto the porch, the screen door's handle is at the right side, but in the following shots the screen door opens the opposite way. (00:10:05)
Continuity mistake: While Lily continues to "light up all the men's lives, except Homer's," when she walks into the power plant with picnic lunches, Homer is wearing a solid peach colored tie, but in the next shot it's a striped peach and pink tie. (00:11:00)
Continuity mistake: When Homer takes Lily home after her date with Burns, as Homer pulls up to her house, the doorknob is on the left side of its front door, but when Lily enters the house, the doorknob is on the opposite side of the door. (00:16:20)
Continuity mistake: At the end, when Lily is back at a pub in England and meets Homer's doppelganger, his single shirt pocket and button are at his right side, but as he and Lily leave the pub arm in arm his shirt pocket and button are on the opposite side. (00:20:20)
Homer Alone - S3-E15
Continuity mistake: When Homer is singing to Maggie, he is standing by the bed. But in the next shot, Homer disappears and reappears.
Continuity mistake: Seymour's time as a POW in Vietnam changes throughout the show. One episode he mentions having been one for a year and a half. Another one he says it was two years. A third one he says it was three years.
Continuity mistake: When Tony Hawk and Homer compete against each other on the skateboards, Bart is on Tony's starting side. When Homer knocks Hawk off his skateboard, Bart is now on the side that Homer started on.
Clown v. Board of Education - S34-E21
Continuity mistake: Fat Tony burns Krusty's tassel on his graduation cap. This causes Krusty to put it out and move it behind his head. In the next shot, it jumps back to the right side of the hat.
The Nightmare After Krustmas - S28-E10
Continuity mistake: When Krusty is in the hospital, the style of his bed rail changes. (00:04:20 - 00:05:35)
The Nightmare After Krustmas - S28-E10
Continuity mistake: When Homer moves Maggie's crib closer to the gnome, the wall outlet under the light switch has nothing plugged in, but in the next shot as Homer exits the bedroom, a nightlight is suddenly plugged in. (00:10:20)
The Nightmare After Krustmas - S28-E10
Continuity mistake: While Krusty is in the hospital talking to Sophie, he pops the IV line from his arm so the tubing hangs down from the fluid bag. But when Krusty tells Sophie that he knows she's reaching out, suddenly Krusty's IV line is properly inserted in the cannula again, only to change back to the tubing hanging down. Also, the white pillow propping up Krusty's foot vanishes and reappears during these shots. (00:05:15)
The Nightmare After Krustmas - S28-E10
Continuity mistake: When Bart opens the door for Krusty and Sophie, Bart is standing on a floor mat, but in the next exterior shot, that indoor floor mat has vanished. Other changes include the cane's handle within Krusty's grip facing the opposite way, as well as Krusty's and Sophie's positions at the outdoor purple mat. (00:06:50)
The Nightmare After Krustmas - S28-E10
Continuity mistake: When Krusty is at the hospital, in the first shot, Krusty's IV tubing is attached to the fluid bag over the bed's safety rail, but in following shots, the tubing is inserted between the rail bars. (00:04:00)
Continuity mistake: After Homer reveals the secret ingredient, one week later, Homer walks into Moe's. In the exterior shot, the door swings open at its right. However, when Moe lights the Flaming Homer, in the next exterior shot, the door swings open at its left. (00:21:00)
Continuity mistake: When Scratchy's head gets sucked into the dam pipe, the back of his head goes in first, but when he is shot out, he is facing forward.
$pringfield (or How I Learned to Stop Worrying and Love Legalized Gambling) - S5-E10
Continuity mistake: When Marge enters the bedroom, the family appears from behind the mattress, and Maggie's outfit is her traditional blue, but when Marge picks her up her outfit is white. (00:17:10)
Continuity mistake: Lionel Hutz runs out of Burns' office leaving both doors open. When Marge says about going, they are closed. (00:20:20)
Continuity mistake: When Homer is talking about Marge getting a job at the dinner table he has a spoon in his hand. Once Marge has finished scowling he doesn't, and there isn't one on the plate either. (00:08:05)
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