So many things to write about, so little time:) So I'll try to keep this short. Most Christians except the fundies will argue that LOTR is a Christian novel, based on the fact that its author was a Catholic, even those who generally dislike the Catholic Church. They will point out the fact that the series is clean, the characters have good morals, women are all chaste and there is no moral relativism of any kind. It also deals with resisting temptation which could be interpreted as resisting the power of sin.
On the other side of the controversy, those of the more fundamentalist bent will say that LOTR promotes so-called white magic as something positive and that various New Age groups love the story even though they are often virulently anti-Christian.
Considering magic, one could say that the positive characters which use it, like Gandalf or Galadriel aren't really mortal (Gandalf for instance, is some sort of an incarnated angelic being) so it's not the same as witchcraft or sorcery but more their innate powers which mere mortals don't possess; though the example of Aragorn calling up the army of the dead is borderline. Yet their second argument deserves more consideration.
While it's quite possible for pagans and Christians to like the same books, isn't it strange that someone like Varg V. (whose Twitter name is Gandalf) who hates and despises Christianity and everything connected with it, is at the same time such a fan of a supposedly Christian story?
In fact, what makes a book explicitly Christian? Characters performing good works or being moral and chaste? You could find these things in other religious traditions, too. What makes Christianity different, is the person of Jesus. “For God so loved the world, that he gave his only begotten Son, that whosoever believeth in him should not perish, but have everlasting life.”
This verse, in its turn, points us to the Genesis and the Fall of Man. Sin brought death into the world and the only way to get saved and have an eternal life is to accept that Jesus died for you personally, for your sins. That is the main tenet of Christianity. With this in mind, let's look at the LOTR universe.
While the narrative of Frodo and the Ring starts somewhere at the end of the Third Age, the Middle Earth and the world around it are much older, and the story of their creation is a subject of another book, Silmarillion, which few have read. Silmarillion describes a distant Creator God Illuvatar who leaves his Creation unfinished and sends his representatives Valar who could be described both as the incarnate angels (but more powerful than Gandalf or Sauron or Saruman who were Maiar) or lesser gods to finish it. Their efforts are undermined by Melkor, the fallen angel whose servant Sauron takes over when Melkor is banished into the utter darkness at the end of the First Age.
Then we learn about the story of the elves who are called the first-born, because Illuvatar created them first. They are immortal and the chief of them have powers close to those of Maiar, like Galadriel. Elves don't die from old age, they can be killed but theoretically could reincarnate as after their death their souls are gathered in a certain place by one of the Valar.
Then the first men wake up. Elves are shocked to learn that they are mortals and that Illuvatar gave them natural death as a gift (it's called the Gift of Men throughout the books) and Valar cannot change this decree. Which fact leads to the fall of Numenor where the ancestors of Aragorn lived, in the Second Age.
The Numenoreans wanted to have an eternal life like elves who were their blood relatives to a degree (long story but Elrond was born of a mortal man and an Elf woman and he was allowed to choose which way to go and chose the immortality of elves while his brother Elros chose the life of men and he was the first Numenorean king) so that they attacked Valinor (the island of Valar) and got destroyed as a result.
While Tolkien himself apparently stated that LOTR is supposed to portray the world many thousand years before Christ, the world he created leaves really no place for Jesus and his sacrifice on the cross. Because, the elves are already basically immortal and have an eternal life, while the death of men is not the last enemy to be defeated as the Scriptures tell us, but rather a Gift of God to them (sounds kinda blasphemous when you write it down like that).
So while I'm not saying that Christians should avoid LOTR (it's still a great book), personally I don't believe that it's a Christian story. In fact, though C. Lewis had suspect theology in many ways, his Narnia books are much more explicitly Christian. Well, that's my opinion and I think I made a good case for it. Feel free to share your thoughts in the comments!