The War of the Simpsons - S2-E20
Revealing mistake: When Ned says "Crème de cassis" while describing what was in the drink he made. There are several animation errors in this shot. His head is disconnected from his body, and his nose disappears. Also, a part of his face is missing between his fingers while he does the 'OK' hand gesture.
Continuity mistake: Milhouse is seen in the school bus during the Lisa segment, even though he was with Bart at the time.
Lisa vs. Malibu Stacy - S5-E14
Revealing mistake: When Lisa says "Now all we need is a name", Maggie's head disappears for a split second right before the shot changes to Bart talking. (00:17:20)
Audio problem: When Homer and Marge are driving to the hospital, Marge says "Let's just say that Bart did it". But her mouth says something else when she says "Bart".
Continuity mistake: When Homer is eating donuts while everyone is doing push-ups, his donut stays the same size every time he bites it.
Children of a Lesser Clod - S12-E20
Audio problem: When the moon says, "You go, Homer,'" the words dont match what the moon says.
The Mook, the Chef, the Wife, and Her Homer - S18-E1
Continuity mistake: Just as Otto is about to put in the grand funk tape you can briefly see an different tape in the recorder. Slow motion is not needed but a sharp eye is.
Audio problem: When Homer is in the asylum talking in his sleep, his mouth stays open when he's talking.
Dude, Where's My Ranch? - S14-E18
Audio problem: When Bart gets the beavers to knock down the tree so that Clara can walk across, she gets onto it and starts walking gently. But when she walks gently, it sounds like she's walking quickly across the trunk.
The Boy Who Knew Too Much - S5-E20
Factual error: Apu (an immigrant from India) could not have been selected for the jury because he is not an American citizen, and this was before he became a naturalized citizen in "Much Apu About Nothing".
Continuity mistake: Before Bart notices the lemon behind the lemon-shaped rock there is actually no lemon there. We can later see that there is too much of a significant gap between these two objects for the lemon to be hiding behind the rock.
Burns' Heir - S5-E18
Continuity mistake: Hans Moleman has brown skin at the beginning of the episode and yellow skin at the end.
Brother From the Same Planet - S4-E14
Continuity mistake: When Homer is showing Pepi the garage, it is purple. In the main titles and in every other episode, it is an orangey colour.
Continuity mistake: Just before Homer (dressed as Krusty) is picked up by Officer Wiggum, his car is brightly painted and has wobbly clown wheels. When he pulls over it's back to the ordinary dented purple car he usually drives.
Continuity mistake: As Homer and Marge are singing in the car, there is one exterior shot of the car, showing it is yellow, but the rest of the scene it is green. (00:05:55)
Continuity mistake: When Homer helps Bart with his racer, the wall of the inside of the garage keeps changing from blue to red. (00:11:45)
Continuity mistake: When Skinner is calmly driving the bus on the way to school, they have obviously gone past a number of power lines (you can see that they have because you can see them in the background). However, when Skinner is trying to get onto the main road, the power lines have disappeared from the background. (00:12:45)
Factual error: In the episode with Donny, when Bart sees principle Skinner go into the school to meet with the rat, the door is backwards. The bars that are pushed in to open a door are always on the inside.
Audio problem: In the beginning, we are at the power plant. We find out that Burns has packed his own lunch, and one thing he packed is a bullion cube. His mouth doesn't move right when he says "Bullion cube".
Tree House of Horror XI - S12-E1
Continuity mistake: In Reaper madness, there's a scene which Homer wakes up being the grim reaper, he puts the robe on at which point he is wearing a singlet. In the next scene where God turns Homer back to his regular self he is wearing his white shirt.
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