Even though it most likely hints at his own backstory, Tramp's falsetto and the way he's animated when he re-enacts a woman yelling for the dog to be quiet can be pretty narmy.
After Boris talked about Tramp's Achilles' Heel being his weakness for females, Lady later tells Tramp, "As far as I'm concerned, you needn't worry about your old heel." Tramp honestly has no idea what that means.
Lady fetches the paper and tears it bringing it through the doggy door. Cut to Jim Dear reading a newspaper with a massive hole in it, even reaching right through to get his coffee. His remark (under the image) finishes it.
From the beaver scene:
Tramp: What you need is a log puller... (The beaver is chewing on the log too loudly to be able to hear him)I said "a LOG PULLER"![voice rumbles] Beaver: I ain't deef, ssssonny. No need to—Did you ssssay 'Log Puller'?"
Then after the beaver frees Lady from the muzzle, err—"Log puller" and attaches it to the trunk. It works a bit too well, and the beaver is flung through the air, still attached to the log, and eventually lands in the water. The log floats over and completes the dam.
Beaver: "S-SS-Say, it works-ss s-ss-swell!"
Lady digging in the garden and accidentally uprooting a tulip. She guiltily props it upright again. Upside-down.
The night the baby is born, Jim Dear is racing up and down the stairs, cheering in his new-dad euphoria. He runs into the doctor, clearly exhausted after being dragged out of bed in the wee hours of the morning and going through the delivery. Jim Dear excitedly shouts "It's a boy! Doctor, it's a boy!" for what is likely the hundredth time. The doctor's response is a wearily deadpan, "Yes... I know."
Tony and Joe preparing Tramp and Lady's romantic dinner. It's about as stereotypical Italian as you can get and it's hilarious.
Tony: Joe! Butch here says-a he wants-a two spaghetti speciale! Heavy on the meats-a-ball! Joe: Tony! Dog's can't-a talk! Tony:[menacingly]HE'S-A TALKIN' TO ME! Joe: Okay, he's-a talkin' to you! You're the boss! [to himself] Mama mia...
Tramp setting up a Let's You and Him Fight between an Irish Cop and a mild-mannered professor to provide a distraction so they can sneak into the zoo. He escalates the argument with a well-timed chomp to the policeman's posterior.
Policeman: AAARRRGGGHHH! Pull a knife on me, will ye?
At the baby shower, the male guests have all gathered with Jim in one room while the women spend time with Darling in another. This isn't what's funny. What's funny is the narmy joke told by one male guest:
"Cheer up, Jim! Old Doc Jones has never lost a father yet!"