Accidental Aesop: Rudolph's Shiny New Year has one about embracing the things that make you unique, but it sort of blurs into "if your physical deformity makes people laugh, let them. Be glad you're making them so happy".
Given that this is the 60s, no way was the message of tolerance intentional, but Hermie's desire to be a dentist unintentionally resembles a Coming-Out Story.
"Rudolph The Red-Nosed Reindeer" originated with a book published in 1939 for a department store's Christmas marketing, which pre-dates the Rankin/Bass special by over twenty years, the song by ten years, and is almost forgotten. The book has little in common with the Rankin & Bass special and focuses only on Rudolph being shunned for his nose before Santa asks him to lead the reindeer team. Hermey the Elf, the Abominable Snowman, Yukon, and Clarice were all created for the Rankin & Bass special & do not appear at all in the book. Ask anyone if they know Rudolph, and they'll either answer with the song or the Rankin & Bass special, and be very surprised if you mention the original book.
The song is another example of this for the book, by the Rankin & Bass animators. Even though it was written in 1949 by Johnny Marks, the brother-in-law of the book's author Robert May, the song was the basic inspiration & theme-music for the Rankin & Bass special, as the animators had only heard of the song.
The trope is averted by the first animated short of Rudolph by Max Fleischer, released in 1948, which predates the Rankin & Bass special and the song. Fleischer's short is based on & true to the original book, though it was later re-released in 1951 with the song added to its soundtrack.
Santa is in fact a big jerk in the original film because he never even "liked" Rudolph (if you can call it that) until he realized that he could exploit his nose. In other words, Santa only cares about Rudolph's nose, not Rudolph.
Rather than being an "abusive" parent, it could be argued that Donner, having grown up in that environment, knew the pain and suffering in store for Rudolph and wanted to spare him.
Did Comet exclude Rudolph from the reindeer games out of prejudice, or was it for the good of the other young reindeer who might be distracted and he chose the wrong words in Rudolph's presence?
Beam Me Up, Scotty!: One line floating around the Internet is Mrs. Claus' demand, "Eat Papa, Eat. Nobody likes a skinny Santa." This exact line is not uttered in the special. She says the former sentence towards the end by itself, and earlier says, "Whoever heard of a skinny Santa?", but never both at the same time.
Darkness-Induced Audience Apathy: One of the major criticisms with this special in modern times is that the majority of the characters are absolutely cruel to both Rudolph and Hermey, including Jolly Old Nick himself, making it difficult to root for most of them.
Ear Worm: Several of the songs. Maybe "We're a Couple of Misfits" especially.
"Silver And Gold", to the point that it's become a holiday standard on its own.
All of the Misfit Toys, not only for being The Woobie, but because most were perfectly good toys, and even the ones that weren't would still delight a child. The particular stand-out is the doll ("A Dolly for Sue"), who had only a few lines outside of the ensemble cast, yet she's beloved for how tragi she is.
The Toy Taker aka. Mr Cuddles seems to be considered one of the better elements of the 2001 sequel.
The Boss Elf, thanks to his memetic "WHY WEREN'T YOU AT ELF PRACTICE?!" line.
Family-Unfriendly Aesop: No one (except your mother and maybe a few others, apparently) will love you for your uniqueness, unless those traits are useful to them. Rudolph would still be an outcast and not a celebrity, if his defining trait could serve others no purpose.
Fanon Discontinuity: The sequel, Rudolph's Shiny New Year, suddenly has a completely different Rudolph puppet, despite starring the same character and taking place immediately after this special.
Not to mention he's also young again in all the sequels.
Or did he get his antlers trimmed once in a while?
Fans of both the traditionally animated 1998 movie and the "originals" will agree that the computer animated Rudolph the Red-Nosed Reindeer & the Island of Misfit Toys never should have happened. Arguably a song or two and using the original designs the other sequels altered are the only saving graces, if even that.
Jerkass Woobie: Donner. He's aloof to his son, and tries to get him to hide the glowing red nose, but when it's revealed, Santa is downright cruel to Donner. And when Rudolph runs away, Donner is the first to go and look for him.
Rudolph was a book long before it became a song, and even longer before the Rankin & Bass special aired. The original book, "Rudolph The Red-Nosed Reindeer" was published in 1939 as Christmas marketing for Montgomery Ward, a US-based department store. The Johnny Marks song didn't come about until a decade later, in 1949, with the Rankin & Bass special not appearing until 1964. The Rankin & Bass version has little to do with the original story, as the animators didn't have a copy of the book & only had the song lyrics to go by.
The Rankin & Bass special wasn't even the first animated telling of the tale. That honor belongs to Max Fleischer in 1948, even before the song was written. The Fleischer short was re-issued in 1951 with the song added to the soundtrack. On top of that, DC Comics issued multiple annuals retelling the story in the late '50s.
Special Effects Failure: Hermey's mouth movements frequently don't match the words he's saying, which at times can be unintentionally hilarious.
Ugly Cute: The Bumble! You can't deny that underneath all that stop motion fur he's kinda cute.
Uncanny Valley: The stop-motion animation has not aged well, and most of the character movements are unsettlingly stiff and jerky.
There were a few sexist lines in the original that were cut in later broadcasts. Among them Donner telling his wife that finding Rudolph was "man's work" and "The best thing to do is get the women back to Christmas Town."
The scene where the reindeer coach prohibits Rudolph from playing reindeer games has come under controversy in The New '10s for promoting a bullying environment.