Bunnymund warns "You don't want to race a rabbit." This is an in-joke to the Aesop fable of the race between the tortoise and the hare - the tortoise won.
Bunnymund opens a rabbit hole and jumps down it. This is a reference to Lewis Carroll's "Alice in Wonderland", where a rabbit hole leads into another dimension.
In one scene, the Tooth Fairy finds a mouse beneath the pillow and identifies it as "one of us, European division". Ratoncito (Little Mouse) Perez or The Tooth Mouse is a children's book character created by Spanish author Luis Coloma in 1894 and that is said to replace lost baby teeth with gifts in Spain, Portugal, France, Italy and Latin America.
At the beginning of the movie, Jack Frost walks through a Colonial American village. In the background, the folk song village musicians are playing is called "Kemp's Jig." This was a popular early 18th-century English dance song named in honor of William Kempe. He was a famous 16th century English comic stage actor whose work influenced modern comedic acting, stand up comedy and improvisational comic skits. He may have performed in some of William Shakespeare's earlier works. He is best known for a stunt where he actually made an entire journey dancing the whole route between London and Norwich (about 100 miles or 161 km).
When Jack Frost is kidnapped to North's HQ, he calls Bunny a kangaroo. Later on, Frost apologizes for that, to what Bunny replies: "It's the accent, isn't it?". This is a reference to Hugh Jackman, who voices Bunny and was born in Australia, known as the land of kangaroos.