The amount of snow on Big Bird when he leaves the roof changes when he walks down the stairs a few moments later.
When the neighborhood kids are singing "Keep Christmas With You" at Bob's apartment, the black boy's cardigan is buttoned and then unbuttoned between shots.
When Gordon grabs Big Bird's hand to pull him into the apartment near the end of the movie, the icicle is on his beak. In the next shot, it is gone.
Bert tells Ernie that he got a soap dish to put Rubber Duckie in so that Rubber Duckie will not fall into the bathtub and sink. Rubber Duckies float.
The wall to the right of Big Bird's nest vibrates when Mr. Snuffleupagus' hindquarters bump into it, giving away that it is painted fabric.
When Oscar screams and groans in pain after Maria angrily pulls him out of his trash can by his neck his mouth doesn't move.
Somehow Bert still has a skate blade on his foot despite having that shoe off.
After Mr. Hooper and Bob wish one another a Happy Hanukkah/Merry Christmas in front of 123 Sesame Street, Mr. Hooper walks off down the street, explaining that he has an errand to run. The gifts he is carrying are the ones he brings to Bert and Ernie (containing Rubber Duckie and Bert's paperclip collection). Bert and Ernie live in the basement apartment of 123 Sesame Street.
A camera tripod is visible in "I Hate Christmas" when Maria stares at Oscar after he throws a snowball at her.
Oscar tells Big Bird, and Big Bird agrees, that if Big Bird can't figure out how Santa gets down the chimney, then Santa will not be able to deliver presents because he won't know how to get into houses. None of the characters, even the grownups, take issue with this reasoning. This ignores the fact that Santa has been delivering presents for generations and, regardless of whether Big Bird (or anyone else) knows how he can get into homes, Santa himself does possess that knowledge, which is all that is required for his gift giving.
When role-playing a scenario on Santa entering/leaving a home, Grover (playing the role of Santa) tells Billy that he cannot get out the back door or the window, because they are locked. Doors and windows lock from the inside. He would be able to unlock the door or window without the need for a key in order to exit.
Maria insists Big Bird would be safe in his nest, even in the snow, due to having an electric blanket and heating pads. Having electric devices such as those, especially of the type available when this special was produced, would be a serious electrocution risk.
When Ernie arrives home he gets the idea to get Bert a cigar box for Christmas and leaves quickly to get it. When he arrives at Hooper's store Ernie somehow has Rubber Ducky with him although he never went into the bathroom at home to get Rubber Ducky before he left for Hooper's Store.