Richard never regains his jacket in the library.
One of the books Richard looks at while inside the dragon is "Alice in Wonderland". The exact name of Lewis Carroll's book is "Alice's Adventures in Wonderland".
During the scene in which the main characters first enter the cave/dragon's mouth, Adventure is overjoyed to find his sword. However, he never lost his sword in previous scenes. After Humpty Dumpty falls, you can see both Richard's glasses and Adventure's sword being carried by the pixies into the cave mouth.
At the end of the movie, Richard's mom takes his glasses and doesn't put them back.
When Adventure got burnt by a dragon and Horror blew out the fire on the mustache, he says "Happy Birthday" but his mouth doesn't move.
During the lighting storm when Richard Tyler (Macaulay Culkin) is riding his bicycle. Immediately, after he exits through the tunnel, you can clearly see the lights used in the lightning gag reflected in his bike windshield.