When Rudolph, Hermey, and Cornelius are floating on the ice, we can see the dogs lying on the sled. When it crashes into the Island of Misfit Toys, the dogs are gone. Obviously, they reappear later in the movie, just before Hermey and Cornelius arrive at the Abominable Snowmonster's cave.
Although the film was copyrighted and released in 1964, the Roman numerals copyright date in the main titles is shown as "MCLXIV", which would be the year 1164, eight centuries earlier. 1964 is MCMLXIV, in Roman numerals.
When the elf is dropping presents from the back of Santa's flying sleigh at the end of the movie, each toy is handed an umbrella for them to use as a parachute. When the elf comes across a toy bird, he pauses and then tosses the bird over without an umbrella, thinking it wouldn't be needed. Unfortunately, that particular bird was the one from the Island of Misfit Toys, and cannot fly (but swims).
According to Sam the Snowman's narration, it was the day before Christmas Eve when Santa Claus initially decided to cancel Christmas due to the bad weather. That same day, however, Santa asks Rudolph to guide his sleigh "tonight" (that is, Christmas Eve).
When Rudolph leaves Cornelius and Hermey in the cabin to set out on his own, the narrator says that Rudolph does this as to not "endanger his friends lives any more." Yet, when he exits the cabin, he leaves the door wide open in supposedly frigid North Pole temperatures.
When Rudolph is in front of King Moonracer he states "When and if we ever get back [to Christmas town] we'll tell Santa". This is in response to Moonracer asking them to get Santa to distribute all the misfit toys. "Dolly for Sue" in response to Charlie's comment about having to wait another year says "But Rudolph promised we'd go this time." Rudolph never made such a promise.
Mrs. Claus keeps prompting Santa to 'Eat, Papa, eat.' But in the final scene, the elf is throwing toys from the sleigh while flying through the night. What difference does it make if Santa is skinny if he never stops to deliver toys?
The train on the Island of Misfit Toys is there because his caboose has square wheels. At the end of the show, however, when the elf is dropping toys from Santa's sleigh, the train no longer has its caboose.
The opening credits are in all capital letters. At one point Burl Ives' name is spelled BURL IVES but later it's BURI IVES. This was corrected in the 50th anniversary collector's edition blu ray/DVD combo.
When Santa leaves at the end, with Rudolph at the head of the team, there are only seven reindeer altogether (including Rudolph himself). There should be nine. The reindeer were divided in a subplot which was cut from the final film.
Sam the Snowman said that when Hermey and Yukon arrive in Christmas Town just as the storm hits, he himself sent them to the cave of the Abominable Snow Monster. But how did Sam know that Rudolph and the other reindeer were in the cave?
When the Bumble and Cornelius fall over the cliff the reindeer look down sadly. If "Bumbles bounce" as Cornelius says later, they would have bounced back up quickly and the reindeer would have seen this.
When Rudolph leaves Yukon and Hermey, he says he hopes Yukon finds lots of tinsel. But Yukon never said he was looking for tinsel. He was looking for gold, then later silver. Only Sam the Snowman mentions tinsel, when he speaking (not singing) between the verses of "Silver and Gold".