The diminutive child robot Tinny Tim is a reference to Tiny Tim from Charles Dickens' A Christmas Carol.
When Fry is in the pet store, one of the pets is Bongo, the one eared rabbit from "Life in Hell", a comic series Matt Groening created and released from 1976 until 2012.
Bender and the robots sing a parody of the song "12 Days of Christmas"
This is the second episode to feature a mummy (the former husbands of the woman that the robots mugged), the first was I, Roommate
The parrot that Fry bought for Leela but got destroyed by Robot Santa was found by Bender to be made as dinner, which he found sometime after his mugging and put inside his container.
The things that Bender and the other robots stole from that woman were:
A pink purse (carried by Bender, probably put it inside his container)
A slipper on a shoe tree (thrown away by Bender)
Three jars of pennies (carried by Tinny Tim, probably lost it or was put inside Bender's container)
Four family photos (carried by the round male robot, probably lost it or was put inside Bender's container)
Two former husbands mummified into tombs (carried by the female robot, probably lost it or was put inside Bender's container)
The gifts Zoidberg, Amy and Hermes give each other are a reference to "The Gift of the Magi" in which a man sells his watch to buy his beloved a comb, while she sells her hair to buy him a watch chain.
The Professor states that Santa invariably judges everyone to be naughty , but at the end he gives Zoidberg a pogo stick for Xmas for being the only good one. It is heavily implied that Zoidberg is the only person to ever be judged as good.
The scene where some kids are skating on ice is a reference to A Charlie Brown Christmas, Bender even sings half of the first lyric before crashing into the pond.