The very beginning of the movie has a rather meta one with the song Circle of Life. If one knows their Swahili and listen to the famous African choruses that start the song, it translates that someone was pointing out to his father that a lion is coming. The father confirms that it is, indeed, a lion.
The part where Zazu starts off singing "Nobody Knows":
Zazu getting crushed by a rhino at the end of "Just Can't Wait to Be King"
Zazu: I beg your pardon, madam, but... GET OFF!!
Earlier in the same song sequence, when Simba and Nala walk through an honor guard-style arrangement of zebras... who then promptly reverse face (literally) and show their butts, tails erect, to Zazu when he tries to follow the cubs.
Timon's absolute confusion when Nala and Simba reunite. After they were at each other's throats. And after she tried to eat Pumbaa, who, after pleasantries, seems to have completely forgotten about it. Which only confuses Timon more!
Simba: Nala, I'd like you to meet Pumbaa. Pumbaa, Nala.
Pumbaa: Pleased to make your acquaintance.
Nala: The pleasure's all mine!
Timon: How do you do.. Whoa! Whoa. Time out.. Lemme get this straight. You know her. She knows you. But she wants to eat him. And everybody's... okay with this? DID I MISS SOMETHING?!
After Timon and Pumbaa meet Rafiki for the first time:
Timon: Hey, what's going on here? Who's the monkey? Nala: Simba's gone back to challenge Scar! Timon: Who? Nala: Scar. Pumbaa: Who's got a scar? Nala: No, no, no. It's his uncle. Timon: The monkey's his uncle? Nala: No! Simba's gone back to challenge his uncle to take his place as King! Timon and Pumbaa: Ohhhhhh...
The start of movie after the "Circle of Life" number which introduces Scar. After Zazu reprimands him for missing Simba's ceremony, he tries to eat him.
Mufasa: (Shows up) Scar! Scar: *With his mouth full* Yes, Mufasa? Mufasa: Drop him. Zazu: (His beak pops out of Scar's mouth) Impeccable timing, your majesty. (Scar spits him out)
In the very same scene above, we get this little dialogue after Scar walks out on Mufasa:
Zazu: There's one in every family sire. Two in mine, actually. And they always manage to ruin special occasions. Mufasa: What am I going to do with him? (meaning Scar) Zazu: He'd make a very handsome throw rug. Mufasa: Zazu! Zazu:And just think, Whenever he gets dirty you can take him out and beat him.
Timon and Pumbaa's verse at the end of "Can You Feel the Love Tonight". Their genuine looking grief, and brusting into tears afterwards, for...nothing.
The Lion King 2
Nuka is always good for a laugh.
"Roasty toasty princess... roasty toasty princess... Hey, is it hot in here, or is that just me? (only now does he realize he's tail is on fire) Fire! Yowww!"
When Rafiki is worried about Kovu, Mufasa (in the form of the wind) implies to him that the solution is to hook Kovu and Kiara up. Rafiki thinks the idea is ridiculous and mocks Mufasa for it—until silenced by the wind. Even when dead, Mufasa has a great sense of humor (and gets his way).
Rafiki: What? Kovu...Kiara...together? This is the plan? (yells) Are you crazy? This will never work! Oh, Mufasa, you been up there too long. Your head is in the clouds! (Wind blows hard at Rafiki) Okay, okay, okay! Okay!
While Kiara and Kovu are helping Timon and Pumbaa scare away the birds eating all the best bugs. Then the group chases after the birds for fun, and promptly run into the birds friends, an entire herd of rhinos.
During the final battle, Timon and Pumbaa find themselves being chased by three enemy lionesses. When they get cornered, Timon cocks Pumbaa's tail like a shotgun and points it at the lionesses.
Timon: Don't anybody move! This thing's loaded. I'll let ya have it! Lionesses:(Run away screaming) Timon:(twirls Pumbaa's tail) Talk about your winds of war.
This exchange before Kovu gets exiled:
Timon: Why I oughta...let me at 'im! Let me at 'im! (to Pumbaa) Here, hold me back.
Pumba: Okay. (holds his tail)
Timon: Let me at 'im! Let me at 'im!
Pumbaa: Okay! (lets go, making Timon lurch forward)
Timon: (Sarcastically) I think you're missing the basic point here.
The Lion King 1 1/2
Funny Moments: When Timon stalls Shenzi for time by proposing. The looks on the faces of Shenzi, Banzai and Ed, and especially Pumbaa, are unforgettable.
Timon: I know what you're thinking, Shenzi Marie. "We're too different. It'll never work. What will the children look like?"
The way he does it is pretty insulting too - "If not now, when? If not me, who?" For believers in the Shenzi/Banzai pairing, Banzai has a pretty smug look on his face the whole time.
Shenzi's overly ridiculous long name (Shenzi Marie Predatorra Veldetta Jackalina Hyena) and after Timon proposes Pumbaa just stands there jaw agape.
Pumbaa asks Timon "Can I be your best man?" Timon's look of "What?!" just increased the hilariousness.
And Banzai's reply.
Banzai: I say we skip the wedding, and go straight to the buffet!
Timon & Pumbaa's MST3King, particularly their reaction to the hula.
There's a game on the DVD called "Who Wants To Be King of the Jungle?". For one of Timon's lifelines, you have the option of calling up his mother, who proceeds to become very excited about talking to the hostess. She calls Uncle Max and says he has to take a picture of her.
Uncle Max: What picture? YOU'RE ON THE PHONE!
Timon and Pumbaa try to break Simba and Nala up unsuccessfully. First time, Timon fires a thorn like a dart gun at Nala, who gets her paw pricked. Simba plucks it out with a grin and tosses it back into the grass...where Timon yelping is heard.
The second one has Timon and Pumbaa drop a spider down to Simba and Nala...only for the spider to see a female spider and fall in love with her. Guess that plan was wrapped up.
The third one has Timon try to throw a beehive at the lions. He throws the hive...but the bees just stay in the air behind him, which leads to him getting stung. Epic Fail indeed.
At some point in "I Just Can't Wait to be King," Simba snatches Zazu (the puppet) away from its puppeteer. The puppeteer shoots an Oh Crap face at the audience and covers himself as if he is naked.
From the 2010 Calgary performance:
Timon: What do you think those stars are?
Pumbaa: I think they're giant balls of gas floating millions of miles away from us
Timon: ...Everything comes back to gas for you, doesn't it?
Zazu gets a lot more snark in this version than he does in the film. In particular, we have this exchange:
Scar: Here I am at the pinnacle and yet the view is bleak. What is wrong with this picture?
Zazu: You're in it, sire?
Another Zazu gem that soars right over the kids' heads:
Scar: Zazu, be serious! I'm in need of some bucking up!
Zazu: Oh, sire, you seem to be bucking up just fine without me!
Rafiki gives what appears to be a very long speech in Swahili. When she's done, she turns to the audience and asks "Do you understand?" After the resounding "no" she rolls her eyes and continues what she was doing.
Even funnier is that since said rant takes place during the making of Simba's portrait and right after the presentation on Pride Rock, it seems very clear Rafiki is talking about Simba and his birth. From some of the expressions and tones she uses, it's very possible part of this rant was about how Simba was conceived (complete with praise of Mufasa's prowess).
During one performance in London, Timon exits to the wrong side. Pumbaa just stands there staring for a few moments, before Timon hurries back on saying, "It's the other way." The entire audience cracked up laughing.