Something very funny happened when Bob Hope filmed this movie, but unfortunately you won't see it *IN* the movie.
In "The Old Grey Mayor", Hope and Lionel Stander play rivals. The shooting script called for them to stand eyeball to eyeball and insult each other. Lionel Stander had one blue eye and one brown eye. When the camera rolled, Bob Hope found himself staring into Stander's mismatched eyeballs, and he couldn't keep a straight face. Hope kept "corpsing" (cracking up), and blowing all his takes. Finally the director had to reblock the scene so that Hope and Stander didn't stand so close to each other.
"The Old Grey Mayor" isn't very funny. What would REALLY be funny is the out-take footage of all those takes when Bob Hope kept laughing at Stander's eyeballs. Hope told this story about his encounter with Stander's eyeballs in his autobiography "Have Tux, Will Travel" (ghost-written by Pete Martin) but he didn't identify the movie in which it occurred.
One of the characters in this movie is identified in the dialogue as "A.K.". This is a mildly ribald Jewish joke. "A.K." is a euphemism for 'alter kocker', an unpleasant Yiddish epithet. That's about as funny as this movie gets.