Oh dear. Many things.
The only guaranteed tears:
The eagles. The very first time I saw Fellowship in the theater, I burst into tears when Gwahir glided into view out of the night shadows, and Gandalf jumped onto his back. And I mean burst into tears. Not a slight welling up, or a tear slipping down my cheek, but blooming burst into tears. I did so on each subsequent theatrical viewing (of which there were a great many). These are good tears, happy tears, tears of power, because I react so strongly to the eagles, I simply have no way to express the reaction other than tearing up. But I often cry at beautiful things. I probably cry more at beautiful things than sad things.
Return of the King... here come the eagles again...
An Unexpected Journey... I completely forgot the eagles were even in this movie when I first saw it, and that first viewing I started crying as soon as the moth appeared and didn’t really stop until everyone got dropped off on the rock.
The Battle of Five Armies... I didn’t remember eagles here either, and I was gone when Azog looked over Thorin’s shoulder and saw them incoming. Oh man, GONE. Completely lost in sobs. There was no moth warning in this one, so it really caught me by surprise.
Why? I have no bloody idea, really. But I bawl at the eagles every time they appear. I will cry sometimes just listening to the scores and reaching that part in the music.
Besides the eagles, let’s see...
Fellowship... not really any other tears shed in here, other than at Gwahir's rescue. I do not cry for Boromir because his is such a good death. I do not cry for good deaths. They are not sad to me. I have occasionally gotten teary when Sam tells Frodo in the boat “Don’t you leave him, Samwise Gamgee... and I don't mean to...”
The Two Towers... Not every time, but I tend to get teary when the elves arrive at Helm’s Deep, and when the ents go to war. My sister had to point out that the music when the ents go to war is the eagles' theme... Ahem.
The Return of the King – Lighting the beacons, when Gandalf takes Pippin with him and Merry watches them ride away, Eowyn’s “ride with me” to Merry, “I can’t carry it, but I can carry you,” eagles, the reforging of Narsil into Anduril. Theoden's speech occasionally gets me. The very end at the Grey Havens when the other hobbits realize Frodo is leaving.
An Unexpected Journey – only the eagles
The Desolation of Smaug – a safe movie! No tears!
The Battle of the Five Armies - eagles, Bilbo’s reaction to Thorin’s death (not Thorin’s death itself, as that is, like Boromir, a good death and is not sad in itself), but Bilbo... oh, Bilbo’s reaction is devastating. Bilbo not being able to say Thorin is his friend to Balin and saying goodbye to the dwarves. Thorin and company busting out of Erebor and Dain yelling “To the king! To the king!” Thorin’s reaction to Fili’s death, that silent mouthing of his name. And in the beginning, Bain appearing with the arrow and The Look he gives his father after he says "this might."