Blooper: Santa's sleigh is supposed to have eight reindeer pulling it (Dasher, Dancer, etc.), which would make 9 total when Rudolph is with them. But when we see the sleigh being pulled at the end, including Rudolph, there are only seven.
Cross-Dressing Voices: The titular character was voiced by Billie Mae Richards, who was ironically credited as Billy Richards from 1965 onward, as mentioned in obituaries mourning her death in September 2010.
Rudolph's Shiny New Year features Red Skelton and Frank Gorshin. This is lampshaded by Skelton, who mentions at one point how red his hair used to be. And you'll spend the whole special waiting for Sir 1023 to start saying "Riddle me this!"
Dark Helmet went from stealing air from a nearby planet to stealing toys from Santa's workshop in The Island of Misfit Toys.