This question requires a pretty big answer... One way you could start gauging its influence on literature is to check out this list of book titles inspired by lines from the Bible, which includes titles like Stranger in a Strange Land, A Scanner Darkly (from "through a glass, darkly"), and East of Eden.
There are also many English idioms that originated in the language of the KJV: [link]
Even a strictly secular reading of a large part of the books of Kings, Chronicles and Judges can prove enlightening when it comes to learning how to plot stories and character arcs. I mean, just look at the stories of David and Solomon, plus the repercussions still felt hundreds of years later as a result of their decisions.
The "wisdom books" - e.g. Proverbs, Ecclesiastes, Psalms - catch a lot of flak for some of their contents, but I'd say that any reader who exercises some discernment as opposed to passing everything in them off as antiquated or silly will learn practical as well as more over-arching thematic lessons.
Also, Victor Hugo had this to say about the book of Job: "Tomorrow, if all literature was to be destroyed and it was left to me to retain one work only, I should save Job."
Though I'm Christian myself, I also try and pull ideas from other kinds of literature, religious or secular. As noted elsewhere in the thread, the writers of the Bible did the same thing. One example that comes to mind is that there are instances in Paul's epistles where he directly quotes Greek literature that would've been known to his audience at the time of his writing.