The Easter Bunny is a character depicted as a rabbit who brings baskets filled with colored eggs, candy, and sometimes toys to the homes of children, and as such shows similarities to Santa Claus. Being a character associated with Easter, he does not often appear in Christmas specials, with the notable exceptions listed below.
Appearances in Christmas specials
The Nightmare Before Christmas
In The Nightmare Before Christmas, Lock, Shock, and Barrel mistakenly abduct the Easter Bunny first when Jack Skellington asks them to capture Santa for him.
Sesame Street

Harvey Fierstein as the Easter Bunny in Elmo Saves Christmas.
The Easter Bunny makes a brief appearance in the Sesame Street special Elmo Saves Christmas, portrayed by Harvey Fierstein.
Earthworm Jim
In the Earthworm Jim episode "For Whom the Jingle Bell Tolls", the Easter Bunny appears as one of the patrons at the Personifications of Abstract Concepts Club who expresses envy towards Santa's popularity.
The Fairly Oddparents

The Easter Bunny as he appears on The Fairly OddParents.
The Easter Bunny is one of the holiday mascots who is featured in the Nickelodeon animated series The Fairly OddParents, appearing as an antagonist in the series' first Christmas episode, "Christmas Every Day!" When Timmy Turner wishes that Christmas would be every day, the Easter Bunny joins forces with Cupid, the April Fool, and Baby New Year in a plot to eliminate Santa Claus and end Christmas. He owns a Dachshund, who he dresses up as a pumpkin-headed dog (claiming it was the "Spirit of Halloween"), since he doesn't like to travel without him. After reaching the North Pole and using Cosmo and Wanda to drain the fairy magic from Santa, the Easter Bunny has it transferred to himself, turning him into a giant. He then prepares to hit Santa with a giant egg-bomb that will blow him to the non-existent date of February 33rd but is stopped when all the other kids in the world with Internet access show up to help Timmy defend Santa. Realizing his and his fellow mascots' error, the Easter Bunny returns the fairy magic to Santa.
Although The Fairly OddParents would eventually get an Easter-themed story (in the storybook Timmy's Eggs-Ray Vision), the Easter Bunny did not appear in it. He did, however, appear at Timmy's Fairy-versary party in the movie Abra-Catastrophe!
The Santa Clause trilogy
The Easter Bunny appears as a supporting character in The Santa Clause 2 and 3, portrayed by Jay Thomas. In these films, he is a member of the Council of Legendary Figures.
Uncle Grandpa
In the Uncle Grandpa "Christmas Special", Uncle Grandpa, who was posing as Santa Claus at the time, is trapped in Sally Smith's dungeon, where she has already imprisoned all the other holiday mascots, including the Easter Bunny.
Mentions in other Christmas specials
In Miracle on 34th Street, when Doris Walker informs Mr. Shellhammer that she fired Kris Kringle from his position as Macy's Santa Claus, he responds, "I don't care if he thinks he's the Easter Bunny; get him back!"
In A Flintstone Christmas, when Fred Flintstone and Barney Rubble meet Santa Claus after he falls off Fred's roof, Barney is willing to believe that it's the real Santa, while a skeptical Fred replies, "Yeah, and I could be the Easter Bunny."
At the end of Christmas Eve on Sesame Street, Oscar the Grouch asks Big Bird if he knows how the Easter Bunny is able to hide all of his eggs in one night.
In the Rugrats episode "The Santa Experience", Angelica Pickles goes over the "history of presents", saying that "even the Easter Bunny started giving them until Santa slapped him with a lawsuit".
In the CatDog episode "A Very CatDog Christmas", when Mr. Sunshine learns that the mall Santa he's been assisting is actually the real Santa, he asks, "Does this mean there's an Easter Bunny?", to which Santa replies, "There'd better be; he owes me sixty bucks."
In the SpongeBob SquarePants episode "Christmas Who?", Squidward pretends to be Santa in order to cheer up SpongeBob. When SpongeBob asks him where his sleigh and reindeer are, Squidward answers, "I loaned them to the Easter Bunny."
In The Adventures of Jimmy Neutron, Boy Genius episode "Holly Jolly Jimmy", Jimmy says that if he can pull off delivering all of Santa's gifts himself, it will prove that Santa isn't real. Sheen responds, "Okay, but the Easter Bunny is real, right?"
In the ChalkZone episode "When Santas Collide", Joe Tabootie finds that the costume company sent him an Easter Bunny costume by mistake.
The Easter Bunny is indirectly referred to at the beginning of Bah, Humduck! A Looney Tunes Christmas; when Bugs Bunny shows up, he mentions how rabbits are normally associated with Easter.
In the T.U.F.F. Puppy episode "A Doomed Christmas", when the villains are brought to jail, Bird Brain gloats, "You may have saved this holiday, but we'll be back to stop the Easter Bunny!", to Snaptrap's confusion.

Santa thinking he's the Easter Bunny in A Fairly Odd Christmas.
In A Fairly Odd Christmas, when Santa gets amnesia, he first thinks that he is the Easter Bunny.
In the SpongeBob SquarePants episode "Sandy's Country Christmas", after knocking himself out on the Cheeks family's giant Christmas tree, a dazed Santa mutters, "Mrs. Claus, the Easter Bunny is here to see you."