Gal of Constant Sorrow - S27-E14
Continuity mistake: When Homer returns from work to get the cat out of the plug socket, he removes the plug socket cover without removing any of the plugs, but when Homer realises that the cat is in the ceiling, both the plug socket cover and the plugs have now been removed from the wall.
How Lisa Got Her Marge Back - S27-E18
Continuity mistake: The pictures of the celebrities behind Lisa and Andrew Rannels change position depending on the whether scene is focusing on Lisa or Andrew.
Continuity mistake: As Homer begins his improv act, Lisa is seated elbow-to-elbow with Dr. Nick. In the long shot view from the rear, she is much further from Dr. Nick.
Continuity mistake: Lenny and Carl's seating in the auditorium changes from being down close to the stage to being higher up and further from the stage.
Continuity mistake: Before Homer begins his speech to the nuclear power plant employees, Lenny and Carl are wearing white/off white shirts with ties. After the speech, Lenny is wearing his usual green shirt with suspenders and Carl his usual gray jacket.
Orange Is the New Yellow - S27-E22
Other mistake: When Chief Wiggum contacts "Rafe Waggum's" parents on his phone, he dials the number and places the phone to his ear, but the phone displays the menu screen and not the call screen.
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)
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: 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 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)
Continuity mistake: After Phil eats the "horse" egg at Moe's he asks to be taken to the hospital, and when it cuts back to Homer at the power plant, the direction of the stripes on Homer's tie switch to the opposite direction, mid shot. The same thing occurs again later. First, when Homer is standing outside the office while Fred talks to Burns about the benefits package, and also when Homer can't stay for Marge's turkey breakfast. (00:07:05 - 00:12:20)
Continuity mistake: While Homer, Carl, and Lenny are at Moe's with their new bud Fred, in the interior shots the swinging front door to Moe's swings open on the left side, but in the exterior shot when they all leave Moe's, that door still swings open on the left side, opposite to the interior shots. (00:08:00)
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.
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.
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