(at around 15 mins) When Homer and Bart are at Krusty Burger, while Krusty is filming his commercial, the pig runs to Homer in fear of being slaughtered. While Homer's looking down at the pig, Bart is shown to be wearing his regular blue shorts instead of the jeans that Flanders gave him.
(at around 46 mins) When Doctor Hibbert asks Mister Burns for power supply for the hospitals, Mister Burns writes it down using his right hand but it is an unchangeable tenet of the The Simpsons that Mister Burns is left-handed.
(at around 35 mins) When they flee the house and jump into the car, the house is on fire, even when Marge runs back into the house to save her and Homer's wedding video, it is on fire. When the townspeople carry the car to the backyard, the house is not on fire.
(at around 4 mins) At the start of the movie the church is shown set back from the street (as it is in The Simpsons), in later scenes (at around 27 mins) it is shown right by the road, next door to Moe's Tavern. On the DVD commentary, the film makers decided it would make more sense to have the church next to the bar since there is a scene where the alcoholics in the bar go pray and the worshipers of the church go drink. But they either forgot or were unable to go back and fix the other scene to make it consistent.
(at around 27 mins) When the dome is being lowered and the guy is thinking whether to stay in or leave, it seems about a foot thick at least. (at around 38 mins) However when Lisa and Colin place their hands on the opposite ends of the dome it seems only about an inch thick.
(at around 23 mins) When Homer dumps the "Pig Crap" silo into Springfield lake, he drives through a load of signs and the concrete fence. When he has dumped the silo and drives away the signs have all gone.
(at around 40 mins) In the Motel scene many times, you can see that the lock chain slot on the door doesn't have the circle where you put the chain through. It has it in a few shots but then in others it is a straight rectangle.
(at around 20 mins) In the assembly scene, Kent Brockman is seen in the first row, he is wearing a white t-shirt, but when the audience spits out the lake water that they drink, you can see Kent to the far left of the close up wearing a red coat.
(at around 48 mins) When the family move to Alaska, the door to their cabin opens to the left (looking at it from the inside) when Homer escapes from Bart and Lisa's "avalanche". But later in the film it opens to the right (from the inside).
(at around 48 mins) When Homer is buried by avalanches the second time and he comes out, his hair is ruffled, when the shot changes to Bart and Lisa, his hair is back to normal, this happens a couple of times.
(at around 29 mins) When Russ Cargill is speaking to the town of Springfield on the big screen, on the top left hand corner you can see the Simpson house with the dome turning on an angle cutting half way through the yard, but in other shots it is perfectly straight right at the back of the yard.
(at around 45 mins) When Marge and Bart are inside the gas station store while Homer puts gas in the pickup truck, Bart places a buzz cola on the counter next to the beef jerky. When Bart aims his sling-shot to shoot the hot dog off Homers' hand, the buzz cola is now near the tip jar. After he shoots the hot dog, the buzz cola is gone.
When Cargill is in the Oval Office, the portraits change. At first, the pictures are of Ronald Reagan and Maria Shriver. In the next shot, the portraits have been replaced by a mountain landscape scene.
(at around 21 mins) When Mayor Quimby calls Cletus over to check the barrier around Springfield Lake, you can briefly see that Cletus has nothing in his hands, then a second later, he's carrying a dead possum.
(at around 9 mins) When Lisa is visiting the neighborhood houses with the petition to stop dumping in Lake Springfield, she comes to Milhouse's house, his lawn is empty but in the next scene there is a large rock in the middle of his lawn.
(at around 5 mins) During the Church scene when Flanders is confessing to the immodest sense of pride; Chief Wiggum is clearly sitting behind Flanders. However, when Grandpa Simpson is doing his monologue of warning, Chief Wiggum seen on the other side of the church.
(at around 27 mins) At one point the church is shown next to Moe's, then later (at around 50 mins) when Barney and the others run out of the AA meeting it's shown with nothing next to it, near the edge of the dome.
(at around 1h 14 mins) When Russ Cargill is going to kill Homer, his clothes are torn up. The last time he is shown onscreen, his clothes were fine. There are no scenes of a previous fight or an animal attack before this. Perhaps he was right outside the dome when the bomb went off, and had less shielding from it than those inside.
When Grampa has his prophetic vision he is upright as he delivers the warnings 'Twisted tail, a thousand eyes, trapped forever!'. However later when Marge watches the footage of Grampa that Comic Book Guy took on his phone he says these words as he is writhing around on the floor.
(at around 29 mins) When Russ Cargill is talking to Springfield for the first time, he says that he trapped them like carrots. You see Moe standing next to Lenny and Carl, who then nod their heads at each other. 6 seconds later, Moe is standing alone at the back of the crowd.
(at around 1h 3 mins) When Homer is riding the wrecking ball, he first bumps into a giant metal statue then into a fork on a bill board. However, in the next shot, he is bouncing between the rock and the "Hard Place" and neither the statue nor the fork are visible.
(at around 57 mins) The face masks in the medicine woman's shelter and the totem poles are not part of Inuit culture. They belong to the Haida Indian tribe who reside much further South down the British Columbia West Coast and the South East area of Alaska in the temperate rain forests.
(at around 56 mins) Marge, Bart, Lisa and Maggie Simpson take a passenger train from Alaska to Seattle, WA. (The exact same train seen in Alaska is shown arriving in Seattle). There is no railroad link between Alaska and any other US state.
(at around 36 mins) When the Simpsons are in Bart's tree house and it is being pulled down by a school bus, we see Grampa Simpson take aim at them with a shotgun that obviously has two barrels, but he cocks it by pumping it. A double barreled shotgun is cocked by pulling the hammer back.
Though Arnold Schwarzenegger is shown being the President of the United States, in actuality he wouldn't be eligible to hold the office, as the US Constitution requires the president to be a natural-born citizen.
(at around 23 mins) Hans Moleman is present when Homer dumps the silo in the lake, but nobody finds out it was Homer until the silo is discovered. But Hans doesn't have good eyesight and isn't very quick on the uptake. There is also a The Simpsons running gag where in several episodes Hans disappears and is presumed dead, only to turn up alive in a later episode, so this could be his disappearance for the course of the movie.
When Homer goes to the fair, he uses the last $10 that the family has to win a truck. In the next scene at a gas station, Homer is filling up gas, and Marge is planning to purchase beef jerky. However, they could have obtained more cash in between scenes.
As the Simpsons probably didn't have their passports, they shouldn't have been able to cross the border to Canada, and again when they reached Alaska. Even if they had they would have been caught and arrested because they were fugitives.
(at around 31 mins) When Homer and Marge go out and look for Maggie, Homer sees his reflection and thinks he's outside of the dome. Yet, Marge's reflection is not seen, even though Homer runs around her.
(at around 19 mins) When Moe dumps the truckload of empty beer bottles into Lake Springfield, look closely at Moe in the truck. As his head turns away, there is no animation of his head turning and it simply "jumps" to him looking forward.
(at around 35 mins) When Marge went back to get one item from their burning house, she took their wedding tape. Homer and Marge had no wedding; they got married in a wedding chapel across the state line.
The goof items below may give away important plot points.
(at around 1h 8 mins) Just as Homer reaches the top of the dome, the next shot looks up showing the people climbing up on the rope to the hole. You see the top of the dome and the hole, but Homer is nowhere to be seen.
(at around 1h 7 mins) When Homer is first seen climbing the dome with super glue on his hands the camera pulls out to show virtually the entire dome, yet there is no sign of the helicopter dangling the bomb through the hole, although a subsequent shot shows it only a few feet above the top of the dome. Nor is there any sign of the bomb itself or the rope that supports it.
(at around 1h 14 mins) Towards the end, Bart and Homer are about to be shot by Russ Cargill, who has a double barreled shotgun, but when he cocks it, we only see one hammer. A double barreled shotgun has two.
(at around 1h 13 mins) While riding a motorcycle around a ball can be done by generating centripetal force, it's physically impossible to ride the motorcycle up to the top of the dome the way homer did.
(at around 57 mins) The medicine woman and Homer don't introduce themselves - since Homer is unconscious when we first see her - yet when she attempts to revive him she calls out his full name. But Homer was probably carrying his driver's license which she read.
(at around 1h 2 mins) When Bart, Lisa, Marge and Maggie are prisoners in the back of the EPA's van, Lisa is playing her saxophone, which she did not take with her when they fled from Springfield to Alaska.