In the movie, Notre Dame is seen with a wide square in front of it. However, during the time the movie is set (15th-16th century) this square did not exist. It was occupied by the Hotel Dieu, a medieval hospital which was not demolished until the 19th century.
When Quasimodo is seen singing "Out There" on the cathedral's roof we can see a set of copper green statues - the 12 apostles - around the central needle. These statues were not added until the 19th century during a general restoration directed by Viollet-le-Duc, a French architect who had read Victor Hugo's novel and was concerned about the possibility of Notre Dame being demolished, as some authorities of the time had considered. Consequently, it would have been impossible to see those statues at the time the movie is set.
Just as Frollo starts scolding Quasimodo for helping Esmeralda escape Notre Dame, he knocks Quasi to the floor. In the next shot, Quasi is seen mouthing something like, "But I...", but no sound is heard.
During the final siege battle, Frollo takes a sword from a soldier, later putting it into his robe on his way up the stairs, when he draws the sword on the roof he draws it from a scabbard never acquired.
After Esmeralda and Phoebus meet, Djali the goat smacks him in the stomach. Esmeralda explains that he doesn't like soldiers. However, throughout the rest of the movie, as well as the sequel, it seems like Djali is female, once Hugo starts taking a shine to Djali.
During the "Out There" number, when Quasimodo sings "If I was in their skin", he is standing over a wall completely in shadow. When he slides down the flying buttress in the very next shot, the wall is brightly lit.
When Esmeralda and Quasimodo are about to jump off the top of Notre Dame, Esmeralda ties a kerchief around Djali's eyes and it wraps around the top of his head. In the next shot, however, the kerchief is wrapped vertically, from top to bottom, and it looks much larger.
When Frollo gets angry with Quasimodo for helping Esmeralda escape Notre Dame he slams the sculpture of her down on the table close to and on the left side of a candle. In the next shot the sculpture is on the right side and is farther away from the candle.
In the scene where Phoebus is arriving in Paris and looking for the Palace of Justice, Achilles sits on one of the soldiers and Phoebus cuts off half of his mustache with his sword. However, in the scene where Quasimodo is being tormented at the festival, the focus goes back to the same soldier. In some scenes he has his entire mustache, in other scenes only half.
In all the scenes with Quasimodo's model village of Paris, we see additional carvings on the model of Notre Dame, especially on the towers. In some scenes, the additional carvings are visible, and in some, they're not.
After Esmeralda unveils the new carvings, she picks up the baker and marvels at it. When she goes to set it down, she sets down the blacksmith. Then when she turns to Quasimodo the sculpture disappears.
When Quasimodo and Frollo are going over his alphabet, Frollo lifts his cup to his lips before saying the letter "F". In the next cut to Quasimodo when he replies "Festival", Frollo's cup is still on the table, despite the audible spit take from Frollo.
When the ferryman demands passage to the gypsies during the attempted escape from Paris in "The Bells of Notre Dame", an arrow pierces his staff. A second later, when the soldiers surround them, the arrow is gone.
When Quasimodo is pulled on stage, Frollo only recognizes him after a person in the crowd identifies him as "the bell-ringer of Notre Dame", even though he should have been able to recognize him immediately as his adoptive son.
When Quasimodo slides down the roof on the "shingle" there are sparks shooting out behind. Only ferrous materials, like iron, will spark. The roof of Notre Dame is made of lead, a metal that will not spark.
The Festival of Fools takes place at Twelfth Night, on January 6 - the song "Topsy Turvy" mentions "the sixth of 'Januervy'". But in a later scene, Frollo brings Quasimodo fresh grapes. Grapes are not ripe in January in France.
At the time of the movie, the use of digital technology to create large crowds of people was fairly new. Thus in many places if you watch the crowds instead of the main characters, you can clearly see how the crowd members all have simple, almost undefined faces and move in very computerized ways. This is especially easy to see at the end of the movie when the little girl pulls Quasimodo into the crowd - watch the crowd that parts for them. Other good examples are just after that when the crowd carries Quasimodo across the square (watch individual crowd members), at the Festival of Fools, and in the shot during the final battle where the gypsies and villagers, moving en masse towards the cathedral, are seen from above.
Esmeralda brings Phoebus to Notre Dame to have Quasimodo hide him with the help of another gypsy, when she and the other gypsy could have taken him directly to the Court of Miracles where Frollo would have no chance of finding him.