The Fall of Elves and Men
The Bible and Lord of the Rings have nothing and everything to do with each other. The latter is a thoroughly pagan story, lacking the metaphors and allegories so many Christian critics reach for. On the other hand, it shows an evident need for Christianity, or at least in the author's view.
Perhaps the clearest way in which Middle-Earth anticipates "something better" (again: Tolkien's view) involves the myth of the fall. "There cannot be any story without a fall," wrote Tolkien (letter 131), and he meant serious business by this remark. There are at least three -- and I think four -- falls we can speak of in The Silmarillion (#2 is controversial).
1. The Fall from Paradise (I) (Elves) The elves fall from Valinor on account of coveting their own Silmarilli gems and refusing to aid the Valar: "They pervert the greater part of their kindred, who rebel against the gods, depart from paradise, and go to make hopeless war upon the Enemy." (letter 131)
2. The Fall from Paradise (II) (Men) The men fall from Eden on account of breaking a mysterious ban. This nowhere appears in The Silmarillion -- for the appearance of the Judeo-Christian myth in Tolkien's own would be "fatal" (letter 131) -- though it is alluded to in the apocryphal "Dialogue between Finrod and Andreth". Men were "born to life everlasting, without any shadow of any end" (Morgoth's Ring, p 314). They were not meant to die originally, but in days forgotten Melkor corrupted this design. This can only refer to the Genesis story, which Tolkien didn't want to use explicitly.
3. Shunning Paradise, Making Paradise (Elves) In the Second Age there is "a second fall or at least error of the elves...they wanted the peace and bliss and perfect memory of the west, and yet to remain on the ordinary earth where their prestige as the highest people was greater than at the bottom of the hierarchy of Valinor". (letter 131) And so the elves created the Three Rings of Power, which enabled them to build pockets of paradise on earth: Rivendell, Lothlorien, and the Grey Havens.
4. Wanting Paradise, Warring on Paradise (Men) Running parallel to the second fall of the elves is the downfall of Numenor, which is about "the inner weakness in men, consequent upon the first fall... achieved by the cunning of Sauron...its central theme is (inevitably, I think, in a story of men), a ban, or prohibition." (letter 131) Men grow dissatisfied with their island of Numenor, and against the ban of the Valar, declare war on them and sail for the Undying Lands, meant for elves and other immortals alone.
In each of the four falls, the children of Eru crave godliness. The men want immortality, and the elves want to be gods of their own creations.
So the mythical backdrop to Tolkien's world is all about "fall". This sets the stage for a pagan epic pointing to a distant future, when fall-damage will be forever healed. Eventually, Eru (God) "will himself enter into Arda (Earth), and heal men and all the marring from the beginning to end" (Morgoth's Ring, p 321).
But for now in Middle-Earth, any trust in the triumph of good is foolish. The pagan Gandalf knows this too well. Under the leaves of Fangorn he reminds his friends that, while he's more powerful White than Grey, "Black is mightier still". Until the era of revelation begins with Abraham, Eru remains remote and distant.
I believe that Tolkien's use of the fall myth in The Silmarillion helps illustrate how he intended Lord of the Rings to "be consonant with Christian belief" (letter 269) without needing to pollute it directly with metaphor and allegory.