Continuity mistake: When Barney becomes his old self again at the training center, he gets on a jetpack while holding a bottle. In the next shot while he is in the air, the bottle is gone.
$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.
Continuity mistake: At the "Do What You Feel" Festival, Homer shows up in his house robe, bear feet styled slippers, unshaven, and unbrushed hair. When he takes the float to save Bart, he's in his regular clothing of blue pants, white shirt, shoes, and clean shaven.
Continuity mistake: The cake that Marge bakes for Homer is initially white, but pink icing instantly appears on top between shots.
Bart Gets an Elephant - S5-E17
Continuity mistake: Bart's elephant crushes the dog house, then it is shown standing, and then it is crushed again.
Lisa vs. Malibu Stacy - S5-E14
Continuity mistake: When the Simpsons are leaving the house to go to the mall. The aerial shot shows that they are leaving in Marge's red station wagon, but the very next shot of them in the car shows that they are in Homer's pink sedan. The shot of the station wagon leaving is probably a re-used shot from a previous episode.
Lisa vs. Malibu Stacy - S5-E14
Continuity mistake: When the Simpsons are about to leave the house to go to the mall, they're all wearing jackets, but when they're in the car just a few seconds later, none of them have their jackets on.
Homer's Barbershop Quartet - S5-E1
Continuity mistake: Homer and Marge's bedroom has photos of Bart and Lisa when they were 10 and 8 years old. However, the story takes place when Bart and Lisa were only 3 and 1.
Continuity mistake: When Ned is saying grace, the plates on the table contain no food, but when Homer comes through the window there's food on the plates.
Continuity mistake: When the robber shoots Apu, he is shot around his leg. In the next shot, the wound is in his chest.
Tree House of Horror IV: The Simpson's Halloween Special IV - S5-E5
Continuity mistake: When Otto hits Hans Moleman's car, it has a number plate on it, which disappears when he crashes.
Tree House of Horror IV: The Simpson's Halloween Special IV - S5-E5
Continuity mistake: Just before Homer is sucked into Hell, he is wearing pyjamas. When he's in Hell, he's wearing his usual day clothes.
Continuity mistake: During the wedding you can see in the background Apu and Sanjay but Sanjay's skin is yellow.
Cape Feare - S5-E2
Continuity mistake: When Bob is working out on the weights, he has a tattoo of something on his knuckles, but when he is watching the exercise video, the tattoo dissappears.
Bart Gets an Elephant - S5-E17
Continuity mistake: Immediately after being chained to a tree, Stampy crushes several objects (a lawnmower, a bike, a doghouse, and a trash can). In the very next shot, which comes after a commercial break, all of these objects are in fine shape. Then the shot changes again and we see them destroyed once again. (00:09:30)
Homer's Barbershop Quartet - S5-E1
Continuity mistake: When Bart and Lisa find the record Melvin and the Squirrels, they hand it to the Comic Book Guy to play. While the record is in his hand you can still see it displayed on the rack.
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