Instead of a passport printed on yellow paper, to make its nature easily identifiable to all, Valjean is given a passport on regular paper (as evidenced by the fact Valjean did not know its nature until reading its text) with the word "Yellow" stamped across it. This makes no sense and is inaccurate to the way the passports of convicts were handled at the time.
After pursuing Valjean and Marius through the sewers, Javert draws his pistol from near his waist and threatens Valjean with it. Javert had just been wading through chest-deep water in the sewers, which would have submerged his gun. A pistol of that era would have flooded with water when submerged, soaking the powder and rendering it unusable until the powder was dried.
After pursuing Valjean and Marius through the sewers, Javert climbs out of a manhole, shrugging the cover up with his shoulders, causing it to spin and flop nearby. A manhole cover of that type weighed nearly 250 lbs., far too heavy for someone to lift as easily as Javert was seen doing.
As part of his prison sentence, Jean Valjean and his fellow convicts are used as galley rowers aboard a ship. France discontinued this practice in 1748, more than half a century before this story takes place. The use of the term "galley" as slang for prison continued in France, hence its use in the original novel upon which this film is based, which may have led to confusion among English speakers when adapting the story.
Gentlemen are seen with the bottom button of their waistcoats unbuttoned. This is a style that came into fashion in the early 20th century. This story takes place in the early 19th century.
Robert mentions sending a shipment to Atlanta, GA. The city of Atlanta was not incorporated until 1847, more than 15 years after the rebellion depicted in Les Miserables. The settlement that eventually became Atlanta was founded in 1836 (4 years after the June Rebellion) as Terminus and was later known as Thrasherville before being renamed Marthasville in 1842, and finally Atlanta 5 years after that.
When the police commissioner hears that two men (Valjean and Robert) are living together, he immediately demands that they be brought in for questioning. The implication is that he believes they are involved in a homosexual relationship. Homosexuality was decriminalized in France in 1791, several decades before this story takes place.
Marius uses the American (mis)pronunciation of "kilometers," pronouncing it "kill-oh-meters." Any country where the metric system is actually in common use pronounces it "kill-ah-meh-tur."