When Cornelia leaves the plate of food outside for Joe the can of Coke is positioned so the word "Coke" is showing. When Joe picks up the plate moments later the can has turned to show the words "Coca-Cola" in cursive on the other side of the can.
Though the action for the 20th Century takes place in New York, all the police vehicles seen are generic black and whites. The New York Police throughout the 1980s were using the distinctive custom painted 'grabber' blue with white roof color scheme, which was something of an icon in itself when New York was portrayed on film during this period.
In the final scene, B.Z. is seen floating off into space and can actually be heard yelling. He obviously would've been dead long before that point.
When Santa talks to Patch in the reindeer stables, Patch shows him designs for his new invention - a clock that wakes people up. A few minutes later the elves are woken up by such a device. Presumably it had been invented before Patch had the idea.
When Cornelia is arguing with one of the students in ballet class, a girl standing behind Cornelia is mouthing the lines of the girl facing Cornelia.
When Patch pulls himself from under Santa's sleigh during their first meeting, he removes the snow to reveal the stage/set underneath.
When Joe and Santa first ride in the sleigh, Santa can be seen speaking an entire sentence as they drive off the roof, yet no words are heard.
Joe does not have a New York accent in every scene.