When the Muppets form a human chain from the skylight, Fozzie Bear is shown sitting on the edge of the skylight with his feet dangling. However, in the very next shot, he is at the bottom of the chain, at eye-level with the crooks.
When Kermit jumps off the bike and is caught on a tree branch, the bike is clearly swerving and about to fall, yet in the next shot, the bike is nowhere to be seen lying on the ground.
In the final sequence, the Muppets are shown in a two-deck airliner; with a passenger section (Statler and Waldorf) and a baggage deck (with the rest). But the plane shown from outside is a single deck parachutist's plane.
The sparklers on Piggy's head in the pool scene change burned lengths.
Gonzo tries to develop a negative of a photo showing the jewel thief but the print is ruined when people open the door and the lights turn on. This shouldn't affect the negative, which he could then just develop at a later time.
When Gonzo, Kermit, and Fozzie are thrown out of the plane, they all land in the same pond, but since the plane was still in flight, they should have been spread far apart. Furthermore, they'd be dead from landing hard on anything from that far up without parachutes.
While obviously humorous, Beauregard would never be a licensed cabbie in London. First off, he wouldn't be driving a American-style taxi, and secondly, to become a cabbie in London, you have to take an exam called The Knowledge, which is one of the hardest exams in the world to take. This is to know where all landmarks are and to be able to, pretty much, be a human GPS (in the days before such things existed to the general public). That way, you know precisely how to get a passenger anywhere in London in the most efficient way possible. Beauregard couldn't even remember how to get to his own place of residence, which was The Happiness Hotel, and needed help from Kermit.
After Kermit walks in on Piggy in Lady Holliday's office, Piggy falls off the desk head first. However, in the next shot, she lands in the trash can rear first. Actually, Piggy does flop off the desk in a way she could easily have landed with her rear-end in the wastebasket.
In the pool dance scene, watching the splashes of water just prior to Miss Piggy rising from the pool with lit sparklers on her head shows that the cut is being shown in reverse.
During the opening musical scene on the city street, all of the cars that are driving around don't have any glass on them; windshields, rear glass, and possibly door windows (though those could be simply rolled down).
In the motorcycle chase, Miss Piggy's "stunt double" - a human being in a jumpsuit, helmet, blonde wig, and Piggy mask - is visible in the long shots.
At the end when the plane's crew member starts throwing the Muppets out one at a time, none of the Muppets he is shown throwing out have parachutes on their backs, then the next shot shows chutes opening up with no Muppets attached to them.
In the DuBonnet Club, a gentleman declines Gonzo's request to take his picture by saying "No thank you, no pictures". At the end of the scene as Gonzo walks away, you can hear the gentleman repeat his previous line but his mouth is not moving.
When Kermit is following Mr. Tarkanian around the news room, a head visibly follows him.
When Kermit is dancing on the bed during "Stepping Out with A Star", the shadow of the puppeteer's arm is visible on the wall behind him.
During "Hey, a Movie", as Nicky Holiday makes his getaway, the studio ceiling lights are clearly reflected on the hood of his car, which is ostensibly driving on a city street.
As Kermit is leaving the dressing room after meeting Miss Piggy and Nicky backstage before the fashion show, you can see the back of Jim Henson in the mirror behind Miss Piggy.
At the Happiness Hotel everyone keeps saying there's no kitchen or food. However, when Beauregard drives through the lobby, the Swedish Chef comes out of the kitchen with food.
When visiting Piggy in the prison, Kermit is referred to as her 'lawyer,' but in the British legal system he should have been called a 'solicitor.'
The voice over the CB radio when Piggy radios in clearly has an American accent.