First, a definition:
From Merriam-Webster OnLine
In this sense, clearly we don’t have the makings of a tragedy. Frodo doesn’t have a conflict with destiny, or the gods, or any other impossible to overcome outside influence, nor is the result of the story (Frodo leaves M-E with the elves and Gandalf, as Men come of age) a disaster, a calamity, or any like description. Not every sad ending is a “tragedy.”
But I’m not sure the ending is “sad.” “Wistful” may be a better term, meaning yearning with melancholy. The ending of the book ends a chapter of “history” that one looks back to with yearning, yearning for the loveliness of the world ages past, the beauty and wisdom of the elves, the sagacity of the ents, the wonder of the wizards, the magic of all of it together. In this it is like any other book of times past viewed with a sense of longing. The difference with this book is that the end of the book tells the passing of that time. All that is associated with the elder days passes symbolically at the end of the book; the fact that the elves take their time to eventually desert the land is irrelevant. In essence, the last chapter of the book could simply say, “And the end of those magic times came, and all who were associated with that magic fled the land of men, and men got about the business of living on their own, in control of the Middle Earth until the Second Music.” For those of us left behind to live in a mundane world, that passing is certainly something that gives us a tinge at least of melancholy.
I can’t put my finger on the passage right now, but Tolkein forshadows the end during the story, when Sam and Frodo have some colloquy about what will happen after the Ring is destroyed, and reference is made to the fact that stories just keep going on, but you can’t expect a grandiose or even “happy” ending. Of course, Frodo simply can’t be expected to live happily ever after; he faced his ultimate test of character and failed. Think about it. Even Galadriel managed to pass this test, so did Gandalf, so even did Bilbo. But Frodo fails. Never mind the reason he failed (the extent and timing of his possession and use of the Ring, the place of his failure, etc.); he will be ultimately haunted by that failure as long as he remains in Middle-Earth. For him, the Blessed Realm is about healing.
In sum, the last chapter is about bridging the gap between the fantasy of the past and the simpleness of the present. And for all who think of the past with a wistful sigh, to lose that past irreconcilably is bound to bring a tear. 
