Here I think I will attempt to clear up a controversy and strike a balance.
Tolkien did not INTEND to write what we might call "Christian" fiction. That is to say that the predominate themes are not meant to be theological in nature. He is not writing an allegory. He in fact detested allegory and criticised C. S. Lewis for his overt use of allegory in THE CHRONICLES OF NARNIA.
Tolkien is writing what he considered myth. He and Lewis both shared a deep love of mythology as a literary genre. Tolkien wrote THE HOBBIT as a children's story. But, like all good faerie tales, it has its origin deep in mythology. THE LORD OF THE RINGS and THE SILMARILLION are the mythology behind that faerie tale. That's why LOTR reads much more like adult fiction rather than TH.
Incidentally, in the movies, Peter Jackson intended for Gandalf's hat to be symbolic of the faerie tale nature inherited from TH. Once the tale turns really serious (at the gates of Moria), Gandalf loses his hat, and the faerie tale is engulfed in the darker mythos of Middle Earth.
So why is it that some believe that TH and LOTR are Christian fiction? Well, Tolkien himself was a devout Christian of the Roman Catholic faith. Much of his own worldview (not theology, per se, but his overall outlook on life) is found in these tales. This should not surprise us. Middle Earth is Tolkien's creation, after all. And part and parcel of Tolkien's worldview are the Christian beliefs of the fallen nature of humanity, the destructive and SEDUCTIVE nature of evil, divine providence, the ideal of self-sacrifice and suffering for the cause of good.
There are some who note similarities between certain characters of LOTR and figures from the Bible. For example, many see in Gandalf a symbol of Christ.
He enters this world as a frail old man (as opposed to a frail infant),
he sacrifices his life to save his friends (and that doing battle with a devil!),
and he is resurrected in a glorified form, full of new power.
Aragorn, likewise, makes a good Christ figure.
He is a long lost king from an ancient bloodline,
the kingdom of his ancestors was divided into Northern and Southern kingdoms (Arnor and Gondor, just like Samaria and Judah),
his true identity was known by his healing abilities (not very evident in the movies, but clearly seen in the books),
and the restoration of his kingdom resulted in a golden age of humanity (as Christ's return is to bring about redemption of the world).
But these similarities between Aragorn and Gandalf and Jesus Christ are not meant to be understood as symbolism, but as TYPOLOGY. Tolkien, a lover of myth, was thoroughly familiar with the way in which similar themes keep cropping up in mythologies. For example, Mithras, King Arthur, the Fisher King, Baldur the Bright, and other mythological figures bear some striking resemblances to Christ. Tolkien actually took this as evidence that the whole world--and not just the Jews--were looking for a Messiah. He regarded such legends as "splintered light", refractions of the truth of Christ that somehow found their way into the different mythologies of the world. The difference? Jesus Christ was that legend entering into human history. Fact and myth are made one and the same in HIM.
So are Gandalf and Aragorn supposed to symbolize Jesus? No. Do they bear striking resemblance to Him? Yes.
Incidentally, Tolkien was directly responsible for C. S. Lewis becoming a Christian. It was through their common love of myth and Tolkien's theory of "splintered light" that "baptized my imagination," as Lewis later put it.
So is Tolkien's works Christian fiction? In the sense that Tolkien's Christian worldview shaped his writings, then yes. But they are not allegory of the Christian faith, like THE CHRONICLES OF NARNIA and PILGRIM'S PROGRESS.
I did not intend to write a doctoral dissertation on the subject, but I hope this will help.