This is based on opinion. Please don't list it on a work's trope example list.
Tear Jerker / Moulin Rouge!
The opening, in which Toulouse sings "Nature Boy", setting the stage for the fact that this movie is going to not end well.
And of course another foreshadowing of the Downer Ending is this line:
Christian (narrating): The woman...I loved... is... [beat] ...dead.
"One Day I'll Fly Away," which shows us that the life of a sparkling diamond isn't as carefree as it seems.
The Ending. Christian's heartbroken, hiccuping cries just add to it.
And also Christian singing the end of the beginning song as the curtain closes and the conductor somberly conducts the orchestra.
The finale version of Come What May may count, upon re-watching; as heartwarming as it is, you notice how much Satine is deteriorating during the final scenes, and knowing what's about to happen afterward...
Roxanne. Just Roxanne. Look at Christian's face.
"The Show Must Go On". We go from Zidler, having confessed the truth to Satine about her illness, thus letting her know she won't ever get her happiness with Christian or as an actress, and convinced her to lie to Christian to save his life. The way he walks away numbly through the backstage, singing about this sort of thing being typical of the sort of life and world they are in, shows clearly how much Zidler actually does care for her and, were the situation otherwise, he'd probably be supporting her and Christian's love. But he can't... Then comes all the backstage people joining in an epic, operatic version of the song. Satine has her own heartbroken moment before resolving to do what Zidler asked her, culminating with her standing dressed to the nines on-stage, but in black, as a Stepford Smiler, with intensely funereal lighting. As if we need any further foreshadowing of what is to come, the crescendo at the climax of the song ends with a long pan up through the Moulin's windmill blades, set to the tune of "Nature Boy" (the song from the beginning, when we first learned Satine was going to die). The whole thing is just so tragic, and the ending so ominous, that even before you see how the movie ends, you're just waiting for the other shoe to drop...
A Deleted Scene showcases how The Moulin Rouge despite being the most popular night club is on the verge of bankruptcy and turning it into a Theater will save it and also achieve Satine's dream of becoming an actress. But to do so, Zidler signed ownership over to The Duke which included Satine making her his. It's also a chance for the other Courtesans (Nini, Fromage, China Doll, Petite) to not have to sell themselves every night as well. He wants her to have her dream and give up being a Courtesan as she's like a daughter to him but knows this is the only way he can do it.
Real Life: Baz Luhrman's (the director's) father, Leonard Luhrman, died in 1999 as production was getting started. The movie payed tribute at the end by saying: "In Memorium: Leonard Luhrman (1934-1999)".
The very end of the credits has pop up on the screen the Arc Words, "This is a story about beauty, freedom, truth, and above all things, love." The very last word is presented with an outlandish, over-the-top, Art Nouveau heart perfectly in keeping with the film's themes, mood, and style, but coming right as it does on the heels of Satine's death and the bittersweetness of the Bohemians' ideals winning even though the Moulin Rouge closes, it's enough to make the chest and throat tight, if not actually bringing tears.