Many well meaning Christians have read into Tolkien's works much more than he intended. For that matter, many well meaning non-Christians have done the same thing, reading into Tolkien's works arguments on various subjects from industrialism and environmentalism to the spread of fascist and communist ideology. Both Christians and non-Christians alike sometimes erroneously refer to the stories of Middle Earth as allegory. But allegory is not the only way to have a deeper level of significance than just the story taken at face value. Symbolism and metaphor are present to varying degrees throughout Tolkien's works. This is a form of typology, and typology is not only present in the Bible but in many great works of literature. Some of it is intended and some not. That which is not is simply a natural outflow of the ingrained worldview of the author. Here is a very good discussion on the typological meaning present in Tolkien's The Lord of the Rings.
So there are the more obvious Christlike typological figures in the story, like Gandalf, who dies fighting a demon which no one else is powerful enough to fight, and ultimately he defeats the demon and is resurrected. And there is Aragorn, the exiled king whom no one recognizes and who waits to defeat the Satanic Sauron and retake his kingdom in the fullness of time. But there are others as well. There is Frodo, who is everyman (or every hobbit), and who volunteers to carry the one object and symbol of commanding and controlling evil to a mount there to be finally destroyed. Frodo's burden, Sauron's ring, get's heavier and harder to bear the longer Frodo carries it and the closer he gets to the end of his journey. Frodo has several moments along the way where he wishes that this cup may pass from him, but ultimately he realizes that it must be him to bear it for it was fated to be so. And there are more, some less obvious but just as real.
Did Tolkien intend for these characters and events to mirror the grand narrative of redemptive history and the battle between light and dark in our world? Perhaps the best answer is that he didn't not intend it. Tolkien was a Christian, after all, and the grand narrative of the Scriptures and indeed of our world was central to his whole outlook and context for everything, including the art of myth making. Tolkien knew that the actual truth about reality in our world was that it was a great war between God and Satan, between the redeeming power of Christ and the damming power of sin, Satan and the curse. This wasn't just a truth he knew in his head but one he held at every level so that when he wrote, even his own internally consistent mythical realm was consistent with the moral and redemptive reality of the world which was made and which is being redeemed by God through the work of Christ.