The costumes, architecture and technology shown in the film are mostly consistent with an 18th-century setting, but Belle recites a poem to Beast ("A Crystal Forest" by William Sharp) which was written in the late 19th century.
During a castle fight scene, Madame Garderobe says 'the fat lady is singing', referring to the saying 'it's not over until the fat lady sings'. But the earliest recording of this is in the late 1800's - BATB is set in the 1740's.
During the song "Gaston" - Gaston is standing on top of the bar and shoots the ceiling causing a lot of debris to fall. The camera then immediately shows a wider shot and there is no debris on the bar, or nearby people, to be found.
As the mob storms the castle, we see them breaking down the main wooden entrance door pushing past Maestro Cadenza. But a few scenes of fighting later, we see the mob breaking down the same door to enter the castle lobby.
During the climax Cadenza loses his 'rotten teeth' which were the keyboards in his harpsichord form as he has very few teeth left when he returns as human but during the reprise of 'Tale old as time' he has a full set of teeth.
Gaston is shown carrying and using a flintlock pistol. An expert with the firearm during speed tests would have trouble reloading one in less than 15 seconds, Gaston manages to do just that in under 10 while engaged in hand to hand combat, outside, at substantial height.
During the end credits of the original theatrical release, two actors are credited with the wrong names. The roles of 'Young Prince' and 'King' (credited to Rudi Goodman and Henry Garrett respectively) were actually performed by the actors Adam Mitchell and Tom Turner.
In the bar scene, during the Gaston musical number, Gaston sings when he hunts he aims his quiver. First, the quiver holds the arrows and really he should be aiming his bow and arrow... but quiver rhymes with shiver so we'll let that pass. The error, however, is while saying this Gaston is actually pretending to aim a rifle or other long gun. Bad mime.
In the opening village "Bonjour" sequence, Belle buys a loaf of bread, places it in her pouch, and a few scenes later it is gone. This is due to a deleted scene where she gives Agathe the loaf of bread.
In "Little Town" Messieur Jean asks Belle where she's going. She answers "to return this book to Pere Robert" but she isn't holding a book. There isn't room for such a large book (shown when she enters the church to return the book) in her apron pocket, especially after she buys a baguette and puts it in her pocket. Such a large book would make a visible bulge in her pocket anyway.
The goof items below may give away important plot points.
When Belle and Maurice are locked in the asylum cart the mob go right behind the cart to get to the castle. When Belle escapes on Philippe, she goes left. She also manages to gain a cardigan/coat between the village and the castle.
We're told at the beginning that many years had passed since the Prince and the palace servants were cursed. Apparently, they were not aging in their Beast / objects forms, respectively, because Chip remained a little boy. However, the villagers were inevitably aging during all these years, meaning that Mr. Potts, Chip's father and Mrs. Potts's husband, must have become significantly older than his wife - actually, it makes no sense that he was still alive and just past his middle age by the time the main action started. The same is true for Mrs. Cogsworth, too.