In particular, the "vessel with the pestle" tongue twister.
"The pellet with the poison's in the vessel with the pestle; the chalice from the palace has the brew that is true!"
Shortly afterwards: "The pellet with the poison is in the vessel with the plasle...the plasle with the plessle, er..."
And then they break the chalice from the palace, and bring out the flagon with the dragon...
The fact that both Griswold and Hawkins fail to remember the rhyme until the chalices are presented — and then Hawkins remembers it, holding the flagon with the dragon aloft triumphantly. Griswold, at that point, just takes the other glass.
The two of them muttering to themselves, trying to remember the rhyme, while Griswold's helmet keeps sticking to Hawkins' magnetized armor.
And the story that for the rest of Danny Kaye's life, people kept asking to do the routine in its entirety with him.
"Yea verily yea!" Especially when they do the whole ceremony in every detail in double time!
The scene where Hawkins seduces Gwendolyn. His impossibly smooth seduction (which occasionally glitches — "I live for a sigh, I die for a kiss!"), his complete theatrical showiness, and the fact that he switches from brave and brazen to cowardly and timid with a snap just make a hilarious scene.
"And a jester unemployed... is nobody's fool!"
Hawkins is echoing whatever's just happened in song because otherwise he has no idea what to do, at which point he learns that he has been seducing the princess. It sounds like he snaps a string, and his voice goes up an octave as soon as he realises what he's saying.