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)
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.
Continuity mistake: When the burglar points the gun at Apu and says "Hand over all your money in a paper bag" Apu immediately replies "Yes, yes, I will". But when Bart sees this incident on TV later, the time between the burglar demanding the money and Apu replying "Yes, yes" is much longer.
Continuity mistake: After Sideshow Bob finishes reading his book and puts it on the table, the pages of the book change from yellow to red between shots.
Moaning Lisa - S1-E6
Continuity mistake: When Bart and Homer first play the video boxing, the window behind them shows that it's broad daylight outside. When Homer blames his loss on Lisa's Sax, it cuts to Lisa's room and it's night all of a sudden.
Life on the Fast Lane (a.k.a. Jacques to Be Wild) - S1-E9
Continuity mistake: When Marge is on the phone to her sisters, her necklace changes from white to red.
The Simpsons Christmas Special: Simpsons Roasting on an Open Fire - S1-E1
Continuity mistake: When Marge reveals the Christmas money from her hair, her necklace changes from red to white. (00:05:30)
Continuity mistake: When the store is being robbed, Homer throws his bag to the side and it lands face down near the stand of chips Homer jumps into. But when they are watching the video of this incident later, the bag is standing upright, far away from the spot it previously was at.
Life on the Fast Lane (a.k.a. Jacques to Be Wild) - S1-E9
Continuity mistake: Near the end when Homer is putting the colored ball things in the tube, right before Marge comes up behind him, the stripes on his tie switch sides for one shot.
Moaning Lisa - S1-E6
Continuity mistake: When Marge says, "Oh Homer, you'll lose your head if it wasn't secured to your neck," her necklace is white. However, when she gives the cup cakes to Homer and Bart, it is red. (00:02:00)
Some Enchanted Evening - S1-E13
Continuity mistake: When Ms Botz is leading Bart, Lisa and Maggie to the living room, the video she is carrying in her hand is brown. In the next shot, it is now blue, and when she is talking to Bart, it is brown again.
Continuity mistake: Before Krusty is arrested, he grabs a drink and there is no olive in it. When he sits down, an olive has appeared in the drink. (00:05:35)
Some Enchanted Evening - S1-E13
Continuity mistake: When Marge and Homer leave for dinner, there is a lipstick mark on the back of the door. The lipstick mark is from the scrapped pilot version ("the Butterfield Incident") in which Homer slams the door on Marge as she tries to kiss him. Some of the animation cells were re-used for the version that ultimately aired on Fox.
Continuity mistake: When explaining RDRR, Ms. Mellon writes "y=r^3/3" on the board behind her. But after she asks, "Don't you get it, Bart?", the equation disappears from the board and she writes it again.
Continuity mistake: Homer draws a face on a punching bag. He starts to show Bart how its done. The punching bag has its eyes filled in now.
Continuity mistake: When Marge kicks Homer out he turns away from the door. When she slams the door he is now stood facing it completely. (00:12:40)
Continuity mistake: At the end, we see Krusty stood next to Mr. Burns. However, as the camera pans over the crowd he has vanished. (00:20:30)
Continuity mistake: Lisa's posture changes at the breakfast table as Bart talks about the school bus. (00:16:10)
Continuity mistake: The position of the street lamp Bart skates into changes between shots. Notice the lines in the pavement. (00:08:20)
Continuity mistake: When the car pulls up by the theatre, you can see it through Bart's window. As the car drives off it's now trees outside his window. (00:08:00)
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