Moaning Lisa - S1-E6
Continuity mistake: When Homer says "I can't concentrate with that infernal racket", the shot goes up onto Lisa's room. From the outside, there is no window frame on the window. But from the inside, there is one going right through the middle.
Margical History Tour - S15-E11
Plot hole: During the Mozart skit, Lisa puts a sleeping tonic in the emperor's wine, but when she drank it she fell asleep also. How could that be if she only poisoned one glass of wine and it was the emperor's?
Continuity mistake: When Judge Harm is yelling at Marge and Homer after they have cut the rope, Marge is wearing lip gloss, and between two shots, it disappears.
Mr. Spritz Goes to Washington - S14-E14
Audio problem: At the end of the episode, when the guy who helped the Simpsons pass the bill says, "I just passed a bill in three hours. Quit complaining," his mouth movements don't match the words.
Continuity mistake: When Mr Burns and Homer first enter the cabin, you can see the door opens with a door knob which is on the right hand side (looking from the outside). Later, after the avalanche, the door seems to open a different way, the knob has been replaced by a handle and it's on the other side.
Continuity mistake: Bart flicks through his chequebook in the car and finishes on The Front page of it having gone through all the cheques. When the shot changes he is holding it open.
Much Apu About Nothing - S7-E23
Continuity mistake: When the mob come to Quimby's office for the first time, the items on his desk change colour and position.
Three Men and a Comic Book - S2-E21
Continuity mistake: When Martin is tied up in the treehouse, at one point Bart takes the brick off the comic book and sets it down near Martin. When Bart and Millhouse fight, the brick disappears.
The Trouble With Trillions - S9-E20
Deliberate mistake: When Homer is being interrogated by the IRS, his package is briefly replaced by a ball of string when the IRS agent sits down in the chair and spins around.
Homer Alone - S3-E15
Audio problem: When Lisa screams "Mom" on the way to school, she just leaves her mouth open and doesn't close it. It should sound like "Maaaaaa" if she keeps her mouth open. (00:03:20)
Homer Badman - S6-E9
Continuity mistake: When Homer's badly-edited "Rock Bottom" interview is shown on TV, the hands on the clock behind him keep jumping around between all the different cuts. But when we saw him giving the actual interview earlier, the clock behind him stayed at 5 o'clock the whole time. The hands on the clock are also slightly different.
Continuity mistake: Throughout the whole of this episode, the picture keeps on changing on Mr. Burns' wall. Sometimes even in the same scene.
Continuity mistake: Right at the beginning, the first shot of Homer and Lisa shows the door open. After Homer says 'in the most wicked condition' and the shot changes the door is closed.
Two Dozen and One Greyhounds - S6-E20
Continuity mistake: There are twenty-six puppies eating from their bowls in the kitchen. As the title suggests, there's meant to be twenty-five.
Continuity mistake: During the Itchy and Scratchy cartoon, things change about the positions of the cats, the clothing, etc. (00:07:55)
Tree House of Horror II: A Simpsons Halloween - S3-E7
Audio problem: In the opening when Marge is warning us about how bad this years episode is, her mouth barely matches what she says.
The Call of the Simpsons - S1-E7
Continuity mistake: When the bear first confronts Maggie, she is on a path. Yet in the next shot when the bear lands on the ground, she is on grass. (00:11:15)
Brake My Wife, Please - S14-E20
Continuity mistake: As Hibbert handcuffs Bart to the radiator he uses the second bar to the right. The next shot it's moved to the next bar to the right.
Simpsons Spin-off Showcase - S8-E24
Audio problem: In Moe's spinoff, when we first hear Grampa's voice, you can see the love tester in the background but the love meter lights do not light up when he speaks.
The Old Man and the Lisa - S8-E21
Continuity mistake: Right at the start when Lisa says "I could still use more paper," the plastic can-rings she was holding vanish in the next 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