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.
When Timon, Pumbaa and Simba lie in the grass looking at the stars.
Timon: They're fireflies. Fireflies that, uh... got stuck up on that big bluish-black thing
Pumbaa: Oh, gee. I always thought they were balls of gas burning billions of miles away.
Timon: Pumbaa, with you, everything's gas.
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.
"Huh, it looks like an oversized shower curtain from Zellersnote or Target or IKEA, depending on where it's performed..."
In the 2011 Montreal performance of the show, Zazu switches 'It's a small world' for 'Vive L'Allouetes!'. It makes sense if you're from Montreal.
In the 2011 Las Vegas show, Zazu belts out another song that starts with a form of "Vive." Three guesses which.
In the 2013 Dallas show Zazu sang the first two lines to "Deep in the Heart of Texas", what makes it even funnier the audience sang along once they knew what the song was.
During the UK Tour, Zazu waits a beat before singing, "And I will walk five hundred miles!...".
The Australia Tour has him singing "Let It Go" which, considering Frozen and the song were all the craze when the show first opened in December 2013, makes Scar shouting "Anything but that!" even more hilarious.
Like with Zazu's song, the dance Timon and Pumbaa perform in order to distract the hyenas varies from place to place. In the UK, they did the River Dance. Cue hysterical laughter from the audience.
Rafiki says 'Tu'as Compris?' at the end of one of her rants, and the subtitles on the screens to the side SHOW IT IN ENGLISH.
One thing which always seems to gain the biggest laugh from the audience is Rafiki swinging in on a vine, telling Nala, Timon and Pumbaa that Simba has returned. What gains the laugh is the fact that after the first swing, she suddenly swings in again, completely unexpectedly, babbling some nonsense. And to top it all off, the priceless "WTF?!" expressions on the trio's faces.
Mufasa (seemingly) fires Zazu. The hornbill's response:
The puppet then attacks its operator ("This is all your fault!"), and Mufasa has to go over to pull him off.
Mufasa: Zazu, it's a joke! *laughs heartily while the puppet and the operator go limp in disbelief*
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.
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.