Continuity mistake: When Homer and Marge are in bed Marge moves up and we can see she isn't wearing anything, but in the close up you can see that she is wearing her pyjamas - note the thing around her neck, then it's back to nothing.
Continuity mistake: After the slinky leaves Moe's hands, and goes out of Maggie's room, we see that Maggie's toy phone is right next to the door way, then when the camera goes towards Moe, and Maggie, then we see the toy phone next to Moe and Maggie.
Plot hole: When Bart prank-calls Moe and tells him his name is Jimbo Jones, Bart says that he lives at 1094 Evergreen Terrace instead of 742 Evergreen Terrace, where Jimbo and Bart are. If Moe had taken those directions, he wouldn't have arrived at the house, instead he would have gone somewhere different. (00:18:30)
The Mook, the Chef, the Wife, and Her Homer - S18-E1
Continuity mistake: When Bart and Fat Tony's accomplices are going to meet Krusty, Krusty's nose can briefly be seen as white when it should be red.
Cape Feare - S5-E2
Continuity mistake: When Bob has finished singing the score of the H.M.S. Pinafore, an English flag comes down. But when he draws his sword to kill Bart, the flag is gone. Where could it have gone?
Continuity mistake: When Barney becomes his old self again at the training center, he gets on a jetpack while holding a bottle. In the next shot while he is in the air, the bottle is gone.
Continuity mistake: When the school bus drops off Bart and Milhouse at the Simpson house, Bart puts his backpack on frontwards and starts krumping. When he walks through The Front door, he is still wearing his backpack on his front. It then cuts to inside the house, and we still see him coming through the door, but the backpack is gone.
Continuity mistake: When the men from Festival Cruise Lines arrive at the Simpson house, they park in the left-hand lane (closest to the house) with the car facing oncoming traffic, seen both when they first pull up and, more, in the background while informing Milhouse about his parents. But in the overhead shot of the house, the car is parked the other way around, following the direction of traffic.
Plot hole: The director of Radioactive Man says to Bart "we found our new Fall Out Boy. That's what I would say if you weren't an inch too short. NEXT!" But Milhouse is the new Fall Out Boy, and he's blatantly smaller than Bart. Bart was standing right next to Milhouse, so the director could see the difference.
Treehouse of Horror XXI - S22-E4
Continuity mistake: In the second segment, after Roger has told them about what happened to him, a radio appears out of thin air.
Treehouse of Horror XXI - S22-E4
Continuity mistake: In the second segment, as Roger is telling them about what happened to him, as he finishes the story of Marge and Homer saving him, in his recount Marge is wearing a shirt and bikini bottoms (the shirt is covering the bikini bra). But when they saved her, she was just wearing a bikini.
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.
Factual error: When Hugh proposes to Lisa using the electronic signage, he spells the word "honor" using the American English spelling. Hugh should have used "honour" as this part of the show was set in England, and Hugh is from England.
Other mistake: During the Planet of the Apes musical final number, as the chorus line dances on stage from left to center with the Statue of Liberty in the background; however, parts of the Statue (the pink background between arm and book) cut off the dancer's head as if it is in the foreground. (00:12:19)
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)
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