Audio problem: Just after Mr Smithers says "...so that we have the energy to from your favourite video game to yummy cotton candy machines", all the children start clapping but the sound of clapping starts BEFORE they start clapping. (00:05:05)
Revealing mistake: During the first ten minutes when the twins are explaining to Bart what incompetence means, one of the twins' body disappears, showing a floating head and hair. (00:06:30)
Factual error: After Bart's prank call with Moe, you can see Bart and Lisa laughing. However the telephone's wire does not connect to the receiver. (00:09:40)
Continuity mistake: In the park, Marge's necklace changes from white to red. (00:15:40)
Continuity mistake: At the very end of the episode, when Homer is being carried off by the crowd, you can see that the crowd carrying Homer, the people are small, so you'll have to look closely, but right before the credits, Marge's necklace changes from red to white, back to red and then finishes with white all in one shot. (00:21:00)
There's No Disgrace Like Home - S1-E4
Continuity mistake: The amount of marshmallows in the jelly at the start changes between shots. (00:02:00)
There's No Disgrace Like Home - S1-E4
Continuity mistake: When they put Maggie down in the Creche she is in the middle of a carpet. Camera cuts and now she's at the edge. (00:04:25)
There's No Disgrace Like Home - S1-E4
Audio problem: When Marge gets drunk at the picnic, she falls on Homer while saying "Homie, did you try the punch?", her lips don't move. (00:06:20)
There's No Disgrace Like Home - S1-E4
Continuity mistake: When Homer walks in on the family eating dinner, Marge has her feet on the second cushion of the sofa. Camera changes and now they are on the third cushion. (00:08:50)
There's No Disgrace Like Home - S1-E4
Continuity mistake: When the Simpsons are spying on the old man, the boy and the old woman, all three of them disappear and then reappear between shots. Whenever there is a wide shot from outside the house, they are not there but when there is a close shot from outside or inside they are there. (00:11:00)
There's No Disgrace Like Home - S1-E4
Continuity mistake: As Homer is carrying the TV towards the pawn shop in the first shot the left window is clear. There is then a tracking shot of the family walking and when they get in front of the store the window has writing on it saying Pawn Shop. The writing is in black and the next shot when they are talking the writing is now dark brown. (00:14:20)
There's No Disgrace Like Home - S1-E4
Continuity mistake: When undergoing shock therapy, the doc raises his hands to his head. They are not present in the next shot. (00:19:00)
There's No Disgrace Like Home - S1-E4
Other mistake: When Homer is at Moe's, there is a boxing match happening on TV. There is a bald guy who is getting considerably battered by the opponent and gets knocked out, and then for some absurd reason he is victoriously standing and smiling as if he had won.
There's No Disgrace Like Home - S1-E4
Revealing mistake: When the Simpsons are eating at the dinner table, Marge's necklace keeps flashing white.
Continuity mistake: When Bart's army is attacking Nelson, there are a lot more kids attacking Nelson than there were training with Bart.
Continuity mistake: At the start, when Nelson takes the cupcakes, you can see that after he has started eating them there are 4 left in the box. Just before he shuts the lid to drop it, you can just see the tops of 6 cupcakes.
Audio problem: When Grandpa Simpson is typewriting a letter in his room, the words he's spelling out loud don't match his fingers hitting the keys.
Factual error: When Bart declares war on Nelson, Herman uses the declaration from the Franco-Prussian war and puts "Bart" in place of "Otto Von Bismarck". The thing is, Bismarck didn't declare the Franco-Prussian War, Napoleon III of France did. A small error, maybe, but something that a war buff like Herman should know.
Continuity mistake: When Bart is training up his army, the colours of the kid's tops keep changing between shots.
Continuity mistake: When Bart enters the tree house after being beaten up, he puts his hat on a peg behind him. In some later shots at an angle you can just see the hat behind him. In the shots from directly in front of Bart, the hat is not on the peg.
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