Factual error: During the song "A Guy Like You," one of the gargoyles trips the trapdoor on Quasimodo's miniature hangman's scaffold. Trapdoors weren't used on scaffolds until the 18th century.
Continuity mistake: In the scene where Quasimodo is reciting his alphabet to Frollo, we see Frollo pick up his goblet for a drink when he says "D?" By "E?", he still has the goblet in his hand. After saying "F?" he raises the goblet and takes a drink. When the camera pans back to Quasimodo saying "Festival", the goblet is in clear sight even when we hear Frollo spit out his drink.
Factual error: Towards the end of the song "Out There" Quasimodo climbs onto the large spire on the roof. That spire was added in the 19th century and thus didn't exist when the movie was set.
Factual error: When Quasimodo and Esmerelda are on the roof of the cathedral, a piece of the roof breaks off and slides with them standing on it. The material creates sparks as it slides. The material would be either copper or (more likely) lead, neither of which can be made to spark.
Factual error: The statues outside of Notre Dame Cathedral are shown as plain stone, as they appear in modern times, although in the story's period (the fifteenth century) they were polychromed.
Factual error: After Phoebus distracts the guards to stop them from chasing Esmeralda, we see the goat smoking a pipe to give the appearance of a beggar. Tobacco (and consequently, smoking pipe) wasn't introduced in Europe for at least another century, as it's a plant native to the Americas.
Continuity mistake: During the song "Hellfire," Frollo pulls out Esmeralda's ribbon and you see him holding it throughout the song. When the monks in the red cloaks appear, the ribbon is gone to appear again moments later. Then when the soldier walks into the room, the ribbon has disappeared for a second time, then when he leaves, it is in plain sight in Frollo's hand again.
Other mistake: During the "Hellfire" sequence, when Frollo is singing "God have mercy on her, God have mercy on me...", his shadow against the wall behind him rises as he backs up closer to the wall. However, when he turns around to press against the wall, his shadow ends up far above his body when it should have become level with his body by that point - like the very next close-up shot of Frollo shows, which is itself a continuity error.
Continuity mistake: During the introduction to Phoebus and Esmeralda, the goat drops a bunch of coins, all through the scene there are no coins on the ground, the coins reappear for Phoebus to through them to Esmeralda (in her old man disguise). (00:17:20 - 00:18:35)