Brake My Wife, Please - S14-E20
Continuity mistake: Whilst Moe is talking to himself about going back to barber college the door to his bar is wide open. When the shot changes to him when he says "Who the hell am I talking to?" the door has suddenly closed. There was not enough time for it to close between shots.
Home Away From Homer - S16-E20
Continuity mistake: For "Home Away From Homer": Bart looks out his bedroom window and sees the college girls doing sexual actions in the Flanders' spare bedroom. But, there is never a window in Bart's room that could possibly face the Flanders' house. [Bart's window in many episodes does face Flander's house but in other episodes like "Bart's Friend Falls in Love" and "Bart the Mother" it faces the treehouse which is the backyard of the house.]
Continuity mistake: After Homer is shot with the cannon for the first time, the cannonball disappears from behind him.
On A Clear Day I Can't See My Sister - S16-E11
Continuity mistake: For "On a Clear Day, I Can't See My Sister": Throughout the episode, the backyard alternates between having a fence at the end and a forest at the end.
Continuity mistake: In one scene Bart is swinging on a washing line then we see Homer talk to Marge and he says "He'll soon get tired of it" then it goes back to Bart and he is facing the opposite direction and he is swinging on two washing lines. (00:04:22)
I, (Annoyed Grunt)-bot - S15-E9
Continuity mistake: At one point in the cycle shop, Homer has his hand in his pocket in one shot, and it's suddenly out in the next one.
Hunka Hunka Burns in Love - S13-E4
Continuity mistake: When they are sitting down in Stu's Disco, there is a little glass next to Mr Burns' glass which vanishes in later shots.
Margical History Tour - S15-E11
Continuity mistake: Lisa empties a glass of red wine that she previously poisoned with sleeping potion. When she falls over sound asleep, her glass is shown on the ground next to spilled wine.
Continuity mistake: When Lisa is explaining to Marge why she is unhappy, the door to her bedroom closes on its own. (00:07:10)
Continuity mistake: When Homer calls the Fatherhood Institute about his poor test results, the guy is holding a cup with 'Super Dad' on it. In the next three shots the cup has jumped to being on his desk. (00:08:50)
Continuity mistake: When the family are on the conveyer belt in the Knowledgeium, Homer turns round and grabs the handrail (after the Troy Maclure announcement) and his fingers are just touching the rail. In the next shot his hands are clasped to the rail.
The Boy Who Knew Too Much - S5-E20
Continuity mistake: When Homer and Bart see each other skipping work/school, Homer is eating a donut then drops it on the pavement when he sees Bart. In the next shot of both of them the donut has disappeared.
Continuity mistake: After Bart's career is dead, he is at his house, and Marge brings out a box of Bart related items. She pulls out a doll and hands it to Bart, as she is handing it to him, the doll's arms disappear, and then reappear when Bart touches it.
Continuity mistake: Ned's clothes are cut and torn in the huricane, but the next day, they are somehow clean and intact. He can't have gone home and changed, otherwise he would have seen everyone repairing his house.
Continuity mistake: When Homer's head gets caught in the middle of the draw bridge, a yellow stripe can be seen down the center of the bridge (where Homer has become stuck). However, in the closeup of Homer being hit by the cars, the stripe has disappeared.
Continuity mistake: When Chief Wiggum and Homer think that the convertible Ruth and Marge are in is a ghost car, they want to hold each other. In this shot, you can see 1 small star out of the backseat of the police car. But in the next shot, from the perspective of the backseat, there is a canyon top where the star would be.
Continuity mistake: When the two chinese guards are dragging Budda Homer along, in an overhead shot we see a desk, with one filing cabinet behind it. In the next shot, the guards stop to rest, and now there are two. (00:17:25)
The Ziff Who Came to Dinner - S15-E14
Continuity mistake: When the Simpsons first see Artie Ziff in the attic, as he stands up, there's a theremin behind him (corresponding with Homer's comment that the "ghost" was playing his theremin). Artie puts a sign up above it. However, when Artie moves to the rafters, the sign he put up is still there, but the theremin isn't.
Continuity mistake: When Patty and Selma bring food to the Simpsons' for one shot Patty's dress turns blue before turning purple again.
Homer vs. Lisa and the 8th Commandment - S2-E13
Continuity mistake: When the family are outside, in the shot just after Homer says "At the worst possible time" you can see the living room - everybody has disappeared in that shot.
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