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.
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.
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)