I do not recommend these books because they are particularly well-written or literary, but Mercedes Lackey's Last Herald Mage trilogy made me cry hardcore as a tween. The main character is a gay teen struggling with his identity on multiple levels and generally going through very difficult times.
I second Lilting Missive's suggestion of LOTR- the latter part of The Fellowship of the Ring makes me cry every time I read it, and I've read it dozens of times.
Watership Down by Richard Adams is also an extremely emotionally powerful book, although I'm not sure it fits perfectly into the fantasy genre.
Another series that I think is really emotional is Garth Nix's Abhorsen trilogy (Sabriel, Lirael, and Abhorsen)- the characters are sacrificing everything to do what's right in these books and they don't always win, which is a contrast to most fantasy on the market.
I also have a real fondness for the characters of Marion Zimmer Bradley's The Mists of Avalon, who are bound by convention and duty to do exactly the opposite of their hearts' desires, although I suggest that you avoid the rest of her work.
If you are interested in other books (non-fantasy) that establish a major emotional connection between reader and the inner world of the novel, I would suggest Jane Austen's work, MIddlesex by Jeffrey Eugenides, Joyce Carol Oates' work, and Cider House Rules by John Irving.