As ancient literature there are indeed parts that are highly unique, especially when one considers the monotheism and the way this is reflected in the poetry sections.
I think what most people expect as literature, the Old Testsment can't really be placed in that section because few have read ancient literary works of the same period.
Jonah is actually one of the best pieces of ancient humor I have ever read, but most people don't know it's a comedy. The flood of Noah has comedic gestures in it meant to tease the defeated Babylonians, but many Christians have obscured Jewish research and our claim that there are redactions we added after the time of Cyrus to make fun of the pagan absurdities in their similar tales. And Psalm 23 is universally recognized for its cadence and simplicity, but again how many have read works from the similar period of other cultures to recognize and appreciate why?
Outside of Jewish works, the only piece from the New Testsment I would consider literary is Luke. He writes the infancy narratives in Septuagint Greek and the rest in Koine as if to offer a flashback, and he puts songs in it. He writes the parables with the flavor seen in Aesop and draws the Passion narrative with the best drama the Gospels have to offer. His Prodigal Son and Rich Man and Lazarus parables are colorfully produced. He is Hellenistic but he is often more accurate than Matthew in his Jewish details.
The problem is that these examples were not intentional. The writers weren't trying to win awards in literature but were catechists and seekers. What JWs and Fundamentalists have done is make claims of Scripture being this and that when their claims have done more to discredit the books in the eyes of logical thinkers more than anything.
My personal opinion is that like any writing you have to take these on their own merit, whatever that might be, and not try to build them up as something they aren't. Only when you do that can you appreciate them for what they really are.