Continuity mistake: When Skinner is calmly driving the bus on the way to school, they have obviously gone past a number of power lines (you can see that they have because you can see them in the background). However, when Skinner is trying to get onto the main road, the power lines have disappeared from the background. (00:12:45)
Continuity mistake: When Homer is sleeping on Bart's school floor, Bart puts his shoe in Homer's mouth. Later, when Homer is screaming about "Cobras," the shoe has been coloured black, rather than the usual blue.
Radio Bart - S3-E13
Continuity mistake: At the start of the scene where Bart talks to Rod and Todd on the radio, their shirt colors switch. (00:08:50)
Continuity mistake: When Homer reaches into the fridge to take out the tub of triple chocolate ice cream, we see it in a brown and white striped tub. When he lifts it out, it's turned into a plain pink tub and has shrunk considerably. (00:03:50)
Continuity mistake: At the beginning of the show, the hospital Mr. Burns is at has a sign in front of it that reads, "Got Tumor?", but when we see the hospital again it isn't there. Additionally, the "Mayo Clinic" sign atop the building gets noticeably smaller in the second shot.
Homer Badman - S6-E9
Continuity mistake: Just after Marge's left jacket pocket bursts open at the candy convention the shot changes to a shot of Marge's face. The next shot her right pocket has suddenly burst as well and the candy in it is no where to be seen.
Homer Badman - S6-E9
Continuity mistake: When one of the pockets of Marge's coat explodes (when Homer is stealing the rare Gummi), a huge pile of candy comes pouring out. When she bends down to pick some up, there isn't nearly as much candy there as before. There are people picking up the candy with her, but they would have to pick it up extremely fast, and they aren't.
Tree House of Horror XII - S13-E1
Continuity mistake: In 'Wiz Kids,' we see Ralph tied up in Lord Montymort's lair. Between a wide shot and closeup, when he says "Dying tickles," his cloak goes from undone to done up. (00:17:05)
The Simpsons Christmas Special: Simpsons Roasting on an Open Fire - S1-E1
Continuity mistake: When Homer has been informed about the Christmas money being spent on Bart getting his tattoo surgically removed, he leaves the house. When he opens and closes the door to leave, the wreath on the door is white with a red bow. When Homer then turns to look his Christmas decorations, the wreath is now green. (00:08:50)
Itchy and Scratchy: The Movie - S4-E6
Continuity mistake: When Bart and Lisa are watching TV, Bart says something like 'I've got my ticket' and pats a ticket that is inside his shirt pocket (which there is none). But later in that same scene when they are still watching TV, both the pocket and the ticket have gone. (00:12:20 - 00:14:15)
Tree House of Horror V - S6-E6
Continuity mistake: When the Simpsons shed their skin at the end, the skin falls to the floor. In the next shot it's gone.
Tree House of Horror XII - S13-E1
Continuity mistake: In the end title sequence, the frogman thing vomits and leaves a stain on the parking lot. When Pierce Brosnan drives away in his car, there is no stain. (00:21:05)
Radio Bart - S3-E13
Continuity mistake: When Homer, Marge and Sting are down the hole and Sting is digging, Homer is wearing a vest. But in the next shot when they find Bart and Homer moves Sting out of the way, he is wearing a shirt. (00:20:50)
Burns Verkaufen der Kraftwerk - S3-E11
Continuity mistake: When Bart starts dreaming what he can do for $5,200, Marge has disappeared from beside him. (00:04:35)
Lisa's Pony - S3-E8
Continuity mistake: After Homer watches the tape, Marge suddenly appears beside him. (00:07:30)
Lisa's Pony - S3-E8
Continuity mistake: Homer's digital watch disappears when he enters the auditorium. It also vanishes in the first shot of Homer in the music store. (00:05:05 - 00:06:25)
Burns Verkaufen der Kraftwerk - S3-E11
Continuity mistake: When Horst looks at the pot full of green slime, his mustache disappears for three separate frames. (00:10:00)
Like Father, Like Klown - S3-E6
Continuity mistake: When Reverend Lovejoy gives Bart a t-shirt "Gabbin' about God", Bart puts it behind his back, where it disappears in the blink of an eye. Look closely (pause helps), but it definitely does disappear. (00:13:40)
Like Father, Like Klown - S3-E6
Continuity mistake: When Bart and Lisa go to see Rev. Lovejoy, the objects on his desk keep disappearing/reappearing. (00:13:05)
Continuity mistake: The old men beside Grandpa when Homer plows the retirement home switch places, depending on whether it's a long or a close shot. (00:10:00)
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