The Burgermeister falling down the stairs because of a toy duck. The choppiness of the animation is the best part of the gag: each of his limbs goes limp one after the other with a significant delay, all the while he has this slack-jawed, bug-eyed look of agony on his face.
The That Reminds Me of a Song that Jessica sings which is completely irrelevant to the plot, but is still funny because of the 1960's acid colors.
And the delightfully cheesy "animation" trick they used to let her look at her reflection in the water; the animators stuck a photograph of her at the bottom of the fountain. Which remains completely stationary even as she sings.
The Running Gag with people criticizing Kris' clothes. When it's brought up a second time, he responds with this:
Meanwhile, the kids are watching the toy in question go up and down with blank expressions.
When the Burgermeister commands the guards to arrest Kris, he comments on Kris' actions as Kris climbs a tree (which the Burgermeister calls it "climbing like a squirrel"), leaps off the roof (called by the Burgermeister as "leaping like a deer") and then sliding down to make a getaway (in which the Burgermeister calls it "slippery as a seal").
"[The South Pole] is on the other end of the Earth — you're just about as lost as you can get."
Grimsley: Herr Burgermeister! Herr Burgermeister! Look what was discovered on your front stoop!
Burgermeister: What? What, Grimsley? The milk? The daily paper?
Grimsley: No, sir. A baby.
Burgermeister: Oh, is that all. (realizes what Grimsley just said, and immediately gasps and chokes) A BABY?!
What really sells it are the gasping and choking noises the Burgermeister makes along with the Spit Take when he hears the news.
Kris' response to the kids worrying about the Burgermeister seeing the toys.