When Belle is bringing her father in the house after finding him in the woods, Phillipe (the horse) is clearly seen eating a mouthful of grass. Since context clues tell us it is winter, all the grass would be dead and unappetizing to a horse.
The band plays the "Bridal Chorus" (better known as "Here Comes the Bride" to most people) composed by Richard Wagner. Wagner composed it sometime in the mid 19th century. The movie takes place sometime around the mid 18th century. The "Bridal Chorus" was not played in places other than Germany until the late 19th century or early 20th century.
During the fighting scene between Gaston and Beast, Gaston says "Belle is mine!" But his mouth is saying something else. This is because, he was originally supposed to say, "Time to die!" but the writer changed it to fit Belle back in the scene.
At the end of the "Gaston" number, women's voices can be heard. But the only women visible are Gaston's three suitors, and their faces are frozen in huge smiles showing off their teeth, so there is no way they can be singing these parts.
When the Beast is fighting off the wolves, there are multiple times when the sound of ripping cloth is heard, implying that the wolves were tearing the Beast's cape. However, once the fight is over and the Beast is seen alone with Belle, there's no damage to his cape at all.
When Mrs. Potts runs over Cogsworth to serve Maurice tea, she, Chip, a sugar cup, and cream cup are on the cart. However, when it stops by the chair, it's just her and Chip. Then, a few shots later, as Maurice is saying hello to Chip, a spoon appears on the tray.
Right before Gaston starts singing "The Mob Song," when he is talking to the villagers, the numbers change from a small number to a large crowd in between shots. Also, they move from far away to up close and all around.
When LeFou is chasing Footstool, he is missing a shoe, which is in the dog's mouth, but when LeFou and his buddies burst through the door, his shoe is on. When the Stove scares them away by flaring up, his shoe is missing again.
When the Beast slams the door to the west wing in anger of Belle's refusal to come down for dinner, a piece of ceiling plaster falls on Lumiere, Cogsworth, and Mrs. Potts, scattering pieces about the place. In the next overhead shot, all of the plaster is gone.
When Belle and the Beast are having breakfast together, before the Beast catches the spoon, his face is dirty. When he catches the spoon his face is clean, and when he tries to eat with the spoon his face gets dirty again. Finally, when he eats right from the plate, as Belle does, his face is clean again.
At the beginning of the movie before Belle starts singing, you can see the steps go down and lead to a path heading to the village. Although when Gaston comes to purpose to her, the steps go down to a path to the barn. Later when Gaston and other villagers come to take Maurice to the asylum it faces straight to the village.
In the song "Gaston" when Gaston sings "when I was a lad I ate four dozen eggs, every morning to help me get large" he grabs the eggs and swallows them whole. Lafou attempts the same to only have them fall on his face. After he says "so I'm roughly the size of a barge!" and shoots his gun a bowl of eggs appears on the counter.
The three blonde girls who love Gaston have three distinct hairstyles in the song "Belle" and during the song "Gaston". They have identical hairstyles in the pull-out shot at the end of the "Gaston" number in the pub.
In the scene where Gaston and LeFou are plotting with the asylum keeper, Gaston elbows LeFou and the mug of beer that Lafou is drinking gets stuck on his face. When LeFou finally gets the mug off, beer splashes around in the cup indicating that the mug is at least half full. However, Lafou turns the mug over on his head to block another blow from Gaston seconds later and the mug is now empty.
During the fighting sequence with Gaston, the Beast's pants get ripped. When the Beast is transformed back into a prince, there is a close-up shot of his foot as it changes. During this moment, his pants are not torn. They go back to being torn when his pants are next seen.
When Maurice is leaving to rescue Belle, a large map gets stuck in the door when he closes it. Two shots later, the map is visible from the inside, but when Gaston kicks the door open, the other side of the map is no longer there as it should be.
Immediately after the ballroom dance scene, Belle and the Beast sit on a balcony and Belle says to the Beast "If only I could see my father again, just for a moment. I miss him so much." and the camera changes to a close up of the Beast. In this close up the Beast's right horn visibly slides around on his head for a moment before snapping into place.
As Maurice leaves the house to rescue Belle, he stuffs a pair of long underwear into his pack. The pair falls out and gets caught in the door when Maurice shuts it. Two shots later, when Gaston and LeFou barge in, the underwear is gone.
After the Beast rescues Belle from the wolves and is brought back to the castle, Belle is trying to mend the Beast's arm. In the shots where she is attempting to place the cloth on his arm and when they start arguing, there are clearly three scratches on his arm, but when it zooms to a closer shot when the Beast has settled down a bit, there are four scratches.
When Gaston is trying to kiss Belle in her house, she opens the door and he falls out into the mud. This would mean the door opens outwards. In the next scene when Belle sets his boots outside (and in all later scenes), the door opens inwards.
When Beast first brings Belle to her room, her bed has a rose in the center at the foot of it. In a later scene when Mrs. Potts meets Belle for the first time and bringing tea, the rose in the bed is gone.
During the opening song when Belle is on her way to the bookshop, the book that she is carrying changes from red, to blue in her basket, to red again. The basket also disappears and reappears at different points in the song.
In the middle of the song "Belle" we see a bird being shot with an arrow and falling to the ground. Lefou picks it up, taking it back to Gaston who is seen holding, not a bow, but a smoking musket which is implied to be what he used to shoot the bird with by Lefou.
When Belle is singing in the opening number by the fountain, we get a glimpse of the book she is reading with an illustration in it as she singing, "It's my favorite part because you see... Here's where she meets Prince Charming...." Later after the song when Gaston grabs her book and flips through it, he says, "How can you read this? There's no pictures in it."
In the "Belle" number, LeFou can be seen carrying a bundle of pelts, one of which has the gray and black ringed tail of a raccoon. But the film is set in France, and raccoons are indigenous to North America.
There are several references to Belle and Maurice being locked in the dungeon but as the Beast correctly states, they were held in the tower. A dungeon would be underground such as a basement. Belle had to climb several flights of steps to get to her father in the tower.
When Cogsworth is showing Belle around the castle for the first time, he asks her to draw her attention to the flying buttresses above. However, flying buttresses are located outside the castle, rather than inside.
When Belle is at the village fountain with the sheep, she points to a picture in her book. Moments later, after the song "Belle" ends, Gaston takes the book and asks how she can read a book with no pictures. However, Gaston does not look through the whole book but only a couple of pages.
Sometime after the opening number, Phillipe abandons Maurice in the woods, sometime before Maurice finds the castle. Later, Phillipe is able to direct Belle to the castle. This makes no sense because he had never been to the castle prior to this. Maurice, on the other hand, returns to the village midway through the film and somehow gets lost trying to return to the castle, even though he had been there before (although after he is left behind by Philippe, he is chased by the wolves the rest of the way). Later, Gaston and his henchmen have no trouble finding the castle, even though they've never been there (Gaston does, however, have the magic mirror). Perhaps the biggest error of all is the fact that none of the villagers seem to have seen or heard of this "hidden" kingdom, which sits deep in the nearby forest.
The goof item below may give away important plot points.
During the final dance with Belle and Adam dancing in the ballroom, the bystanders on the outskirts of the dance floor are dressed in medieval clothing and are drawn in a distinctly different style from the characters in the movie (except for the inserted servant staff). This is because the film re-used crowd cells from Sleeping Beauty (set in the Middle Ages) in this scene. As such, there are several centuries of difference between the characters of the film and the crowd surrounding them at the end.