What "non-religious purpose" would you want?
I'd say most atheists aren't going to scream at the mere notion of the Bible. But it's all about context. Are you just going to read Genesis? Are you going to RiffTrax the Bible, making fun of it? Or are you going to just read the cool part of Revelations?
Merely reading the Bible to anyone in the West is unlikely to do very much. Biblical imagery is deeply ensconced in our society: While there's a lot of inaccurate information and myths out there about the Bible, even those are based on cultural interaction with and remixing of the text. Even from an excitement perspective, you actually need to go kind of deep to get people interested. The Nephilim, Shedim, etc. are pretty cool and relatively unknown, so that'd be fun.
I wrote an opinion assuming you might want to excite someone on a religious basis. I'll leave it in for completion:
Because of the iconic nature of Biblical imagery, if you find the Bible compelling, that's fine, but don't expect many people outside of your faith tradition who have relatively certain beliefs to be swayed.
I have a friend who, in elementary and middle school, suddenly became an evangelical Christian because he had a dream about the four Horsemen. He insisted he hadn't read those passages in the Bible, but even at the time I pointed out that the Four Horsemen are such a cultural fixture that the idea that he hadn't heard of them even once was just ludicrous. The principle of parsimony made clear that it was far more likely that he had absorbed the idea through cultural osmosis than anything else.
Today, he's a relatively left-leaning fireman in the Pacific Northwest whose spirituality is very clearly non-denominational and skeptical.
Even if you have a big (literal) "Come to Jesus" moment just from reading the Bible, it may be short-lived.
Worse, the Bible is actually really boring, and there are glaring contradictions in it. I know, many Christians (and Jews and Muslims) believe there are interpretations that can rectify the apparent contradictions. It's noteworthy that there are in fact many such interpretations: dispensationalism, for example, is just one form of supersessionism, and while most Christian groups are supersessionist to some degree, not all are.
If you really want to try to engage with people from different faith backgrounds, try this:
If you're going to do this, of course, recognize that if you're going to ask others to engage with you, you should engage with them. If you really feel someone will benefit from reading Left Behind, you should be prepared to read God Is Not Great.
- Pick a few passages that really speak to you. Try to communicate what you think it says in a way that shows what moves you about it.
- Listen to some of their criticisms of parts of the text. Find some context and come back to them with a broader context or interpretation. At the very worst, you'll deepen your own faith and understanding!
- Suggest books that might help them see the text better. Try to avoid really obviously propagandistic texts like Heaven Is For Real or the Left Behind series: there are some good texts with a Christian ethos, like Chronicles of Narnia or The Lord of the Rings. Those latter two are too well-known, but you can probably find less well-known ones that might be a good opening.