When Jane gets caught on a vine during the chase, Tarzan is swinging away then he sees he's lost her and his look of absolute exasperation is priceless. It's like, "Oh, for the love of..."
Seriously, the stuff Tarzan puts himself through to rescue this strange shrieking creature is both amazing and side-splittingly hilarious. His face when he does the splits...
Tarzan returning the drawing to the angry baboon parent while speaking to it in its language.
When Tarzan unknowingly gets a little too personal (read: trying to see what's underneath Jane's skirt), Jane's sudden pitch change, delivery and kick to his face is just perfect.
In the sequel, Terk does an amazing Kerchak impersonation.
Jane's entire rant about her first encounter with Tarzan:
"And Daddy... they took my boot!"
"And I was saved! I was saved by a flying, wild man in a loincloth!"
What makes it even better? Most of that speech was ad-libbed.
There's also how hilariously excited Jane's father gets when she tells him about Tarzan. When she mentions how Tarzan moves around with his hands on the ground, he comments that it sounds like how Jane's aunt walks around.
When Clayton tries to teach Tarzan English by drawing on the chalkboard and he gets frustrated:
Clayton: (shaking his fist at Tarzan) No! No, no, no, no, no!
Tarzan: (grabs chalk out of Clayton's hand and shakes his fist back): No! No, no, no, no, no!
And when he asks Tarzan where the gorillas are.
Clayton: Where are the gorillas? (Tarzan tries to examine, but Clayton grabs him by his wrist) GO-RIL-LAS! Tarzan:(imitates him) GO-RIL-LAS!
And before that where the herd of elephants are chatting amicably about piranha being native to South America, busily Leaning on the Fourth Wall.
Tantor's Mother: For the last time, honey, there are no piranhas in...
Other Elephant: MY BUTT!!!
Baby Tantor's entire introduction is pretty funny. Squeaky baby voice while being a phobic elephant is comedy gold.
Tantor: MOM, are you sure this water's sanitary? It looks questionable to me...Yuck! But what about bacteria?
Baby Tarzan and Kala's first meeting. Just try and watch their reactions without laughing.
Tarzan repeating everything Jane says. "No, no, no, no, *clears throat* I'm Jane."
Or this bit...
Jane: (As Tarzan grabs her to go vine swinging again) Ummm...uh...can't we WAAAAAAAAAAALK!
Tarzan: WE WAAAAAAAAAAAAAALK!
Terk in Jane's dress.
"I'M GONNA KILL HIIIIMMMMMMMM!"
"Actually, I think that dress is very slimming." "Really? I thought it was a little revealing."
Tantor freaking out at the campsite, capping off with a pan to tableware resembling Mrs. Potts and Chip.
Terk: These things aren't alive.
Tantor: I knew that.
When Tantor forces Terk to join him in rescuing Tarzan, and sets off for the ship by leaping straight off of a cliff and into the ocean.
Tantor: (thrilled) I NEVER FELT SO ALIVE!"
Terk: (pissed) GOOD! Because I'm gonna kill you!
This is followed up when we see Tantor climb on the ship and start beating up Clayton's men. Cut to below deck, where Tarzan, Jane, Jane's father Professor Porter, and the ships captain are all being held captive. Tantor's foot goes through the deck and stops just above Professor Porter's head, as Tantor gives a trumpet. Cue the professor completely missing the massive foot above his head and commenting "That sounded just like an elephant!"
And when the Professor punches the ship's wall just as Tantor's weight tips the ship and sends everyone slipping to the side. The Professor then remarks "Don't know my own strength".
"Trashin' The Camp", aka "Do Bop, She Do".
The funniest part is that during the musical number, we cut to Kerchak and the herd, then to the professor and Clayton as they faintly hear the music and glass breaking.
Professor Porter swinging on a vine with his own loincloth and imitating Tarzan's yell at the end.
Tarzan: (trying to sneak up on Kala) (winds up an elephant trumpet)
Kala: Don't even think about it.
Tarzan: (splutters out a deflated trumpet and swings around in front of her) How'd you know it was me?
Kala: I'm your mother, I know everything. Now where have you been?
Tarzan: (chuckling) I thought you knew everything.