In the Hobbit, when Bilbo invites the remaining dwarves to tea at 4:00, but they can come anytime - the door is always open.
I do tear up when Merry asks Pippin if he is going to die. Another thing that gets me is when Pippin tries to get Gandalf to help with Merry, but Gandalf is too busy... until he finds out it is Merry, then he drops everything and comes.
The love the fellowship have for each other is splendid. Legalos taking Gimli to the Grey Havens. I also cry when Frodo must leave Sam... "I am wounded, Sam."
These are great books, but reading the appendix is really finishing the story, and when you find out that love and devotion continued even until death... it touches you.