Fair for Its Day: Max Perry Mueller argues in Race and the Making of the Mormon People that the Book of Mormon actually challenged some aspects of the 19th-century racial worldview by preaching against racial schisms and that everyone could be saved regardless of their race (2 Nephi 26:33). However, other passages haven't aged so well as they seem to indicate that one could become "white" through conversion (though this may be taken as figurative).
Hilarious in Hindsight: The Lamanites utterly fail to take the city of Noah, and the narrative describes how some of the Nephites were only wounded due to arrows striking their unarmored legs. No doubt more than one of them took an arrow in the knee.
Tearjerker: The final chapter, when Moroni is the Last of His Kind. All the time he is writing, he is completely alone, his father has been killed, and he's doing all he can do to finish his work while on the run. He knows full well that God will quit protecting him after he finishes, and he wants it to be this way, as he has nothing else to live for. The last verse is him saying goodbye to the reader and hoping to see them in the afterlife.
You'll notice that there's no cursing for the first twenty minutes or so of the play until they finally arrive in Africa during "Hasa Diga Eebowai." That's to lull the viewer into forgetting that they're watching a show written by the creators of South Park, and then club you over the head with as much cursing as humanly possible.
Hatimbi mentions during the song that "80% of us have AIDS; Hasa Diga Eebowai!" He later rattles off a list of people he knows that have AIDS, including the butcher, the teacher, the doctor, and himself, except for his daughter who has "a wonderful disposition." If you're keeping track, that's four out of five people, or 80%.
"Joseph Smith: American Moses" Crosses the Line Twice in every act of the play, but almost every message is something directly applicable to the Ugandans: stopping rape, preventing genital mutilation, and promoting clean water. The only thing that stands out is their literal interpretation of Go Forth and Multiply, which is probably the most grotesque part of the play. Except we never see Elder Cunningham talk about that aspect of the scripture (and his own anxiety in "Baptize Me" makes it seem unlikely he'd give a sermon on orgies with Jesus). The person who does talk about that part of scripture? Elder Price, in "All-American Prophet," as a lead-in to talking about himself. It seems like Arnold isn't the only one to blame for that racy number.
Fridge Horror: Elder McKinley's situation, when you remember the Ugandan government's general attitude towards homosexuality.
"Funny Aneurysm" Moment: Elder Price's dream of going to Orlando has become this, following the deadliest mass shooting in US history taking place in the city in June 2016. Worse yet is the fact that said shooting took place in a gay nightclub. Andrew Rannells, the original Elder Price, is gay, and Price himself could be considered Ambiguously Gay.
Harsher in Hindsight: Elder Price's Dark Reprise of the "Orlando" refrain can hit a little too close to home for some after the June 2016 mass shooting. See above.
Depending on the production, Elder McKinley and Elder Price. It helps that McKinley is canonically gay and Price has no love interest in-story (plus Price's original actor, former 4Kids voice actor Andrew Rannells, is gay).
In some later productions, after returning from the village covered in the blood of a recently executed Ugandan, Elder Price grabs Elder McKinley by the arms and/or face in excitement. Elder McKinley almost tries to kiss him.
Some versions have them make out during "Spooky Mormon Hell Dream".
Elders Price and Cunningham have quite a bit of this as well.
It also helps that the musical never portrays Mormons or even Mormonism itself as bad. The whole point of the musical is that religion can be used to bring people together, and that's what the Mormons ultimately do.
During the touring production, not only has the LDS church bought multiple ads in the Playbill, but they station missionaries at the entrance to hand out cards. Protesting isn't going to work, so why not use it to their advantage?
Playbill ad: Our version is sliiiightly different.
Nightmare Retardant: "Spooky Mormon Hell Dream" is deliberately anything but spooky, which is part of the reason why the number is so hysterical.
Portmanteau Couple Name: McPriceley (Price/McKinley), Arnaba (Cunningham/Nabulungi), and Pricingham (Price/Cunningham). Also 'Churchtarts' and 'Schravis', as detailed below.
Ships That Pass in the Night: 'Churchtarts', the pairing of Elder Church and Elder 'Poptarts' Thomas, who each have a verse in "Turn It Off". They have about ten lines between them, and never speak to each other.
Even more random is 'Schravis' - Elders Schrader and Davis. The former isn't even named outside of the script.
Unfortunate Implications: This article brings up a few, albeit by missing the point of the show. Also, the LDS Church's unkind opinion on homosexuality renders McKinley's situation rather uncomfortable, especially when the dialogue at times seems to support the "moral validity" of his repression. It also doesn't help that the message of the musical is that Mormonism does teach good moral lessons, implying (obviously accidentally, considering Stone's and Parker's well-established acceptance of gay people) that homophobia is one of those "good moral lessons," or is otherwise just one of the silly, unimportant side-effects of the Mormon teachings like the other ridiculous factual teachings made fun of throughout the show.