Continuity mistake: When Bart runs his nails down the chalkboard, the image on the board changes slightly; note the legs. There is a gap between one leg and the body in one shot, but not the other. (00:14:00)
Continuity mistake: When reading his responses to college applications, Homer pushes his plate away, but as he stands up, an empty plate appears in front of him. (00:07:00)
Continuity mistake: Homer puts his photo on his application, and there is no white spot behind his head in the picture. When the shot changes, there is now a white patch in the photo. (00:06:00)
Continuity mistake: The writing on the entrance essay Lisa is reading changes between shots. (00:05:40)
Continuity mistake: Burns brings out the mystery box and places it on the desk. This box then simply disappears. (00:03:40)
Cape Feare - S5-E2
Continuity mistake: The design of the wall socket in Itchy and Scratchy changes between shots. Note the screws holding the cover. (00:03:40)
Cape Feare - S5-E2
Continuity mistake: At the very start, McBain walks out on stage and there is no gold star on the stage. After he says hello, he walks and stands on a gold star. (00:01:50)
Cape Feare - S5-E2
Continuity mistake: When captured at the end, Bob says, "By Lucifer's beard," and grabs hold of the railings. When the shot changes, he's back to hanging over the railings. (00:20:20)
Cape Feare - S5-E2
Continuity mistake: When Bob is laying in the public street, note the position of Bart's items he dropped. The strap moves, and the position of the apple in relation to the edge of the sidewalk slab changes. (00:16:00)
Cape Feare - S5-E2
Continuity mistake: When the Simpsons arrive at Terror Lake, the car is parked in the middle of the road away from hedges. When Bob is standing on the rake, the car has now moved next to a hedge without being driven. (00:14:50)
Cape Feare - S5-E2
Continuity mistake: When the Simpsons arrive in Terror Lake, Homer has his arms around Bart and Lisa. However, when the shot changes to behind, his hands are on his hips. (00:14:40)
Cape Feare - S5-E2
Revealing mistake: After Bob has announced who will be safe, Homer runs into Bart's room and, as the door opens, it goes through a baseball bat on the floor. (00:11:40)
Homer's Barbershop Quartet - S5-E1
Continuity mistake: The display of Homer's control desk changes between the shots of the chicken at the work station. (00:19:20)
Homer's Barbershop Quartet - S5-E1
Continuity mistake: When Homer wipes the dust off the box, there is still some visible dust on the box. When the shot changes, this dust has disappeared. (00:16:40)
Homer's Barbershop Quartet - S5-E1
Other mistake: When Grampa is listening to Baby on Board on the radio, the top right of the screen has a purple rectangle flash on screen for 1 or 2 frames. (00:12:30)
Homer's Barbershop Quartet - S5-E1
Continuity mistake: The wording "Be Sharps" changes size on the album when the family is laughing at Bart. It gets much smaller. (00:04:00)
Homer's Barbershop Quartet - S5-E1
Continuity mistake: Skinner puts the helmet on his head and we see Hermann from inside the helmet, and he has no cigarette in his mouth. When we see the shot from outside, he does. (00:03:10)
Revealing mistake: When the dog attacks Chief Wiggum, you can see a floating flashlight, a person in the bottom right corner missing an eye, and a cameraman pointing the camera elsewhere instead of at the ground. (00:04:54)
$pringfield (or How I Learned to Stop Worrying and Love Legalized Gambling) - S5-E10
Continuity mistake: When Marge enters the bedroom, the family appears from behind the mattress, and Maggie's outfit is her traditional blue, but when Marge picks her up her outfit is white. (00:17:10)
Continuity mistake: Apu grabs the ham and crosses out of the expiration date with a thick black marker. When he places the ham on the shelf the black marker cross out is gone and the writing on the pack looks perfectly printed as before.
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