I think it's fine. I'm a Christian attending a graduate theology school and it wouldn't bother me if an atheist used a biblical name. I would probably assume someone who used a more obscure biblical name is Christian/Jewish, however. I find that such names are most often used by conservative evangelical Christian or (in the case of Hebrew names) Jewish families.
If you like a name, go for it. The Bible is a foundational piece of Western literary/cultural heritage and has significance even if you aren't religious.
I am Christian (Catholic) and I would not be offended
I really think it depends on how closely they're associated as well as if there are other well known namesakes.
So a name like Abraham, while it's very associated with Judaism, is also held by some very famous namesakes. So if I met someone named Abe, I'd be just as likely to assume their parents admired Abraham Lincoln as to assume they were deeply religious.
I'm not really sure where the line is, but I certainly wouldn't be offended in any way, except maybe if someone used Judas. Like others have said, it's more an issue of assuming you're religious because of Malachi or Zachariah. But names like Esther, Ruth, Lydia, Felix, and others....weren't even originally part of Judaism or Christianity--they were from other nations, so just because they happen to be in the Bible doesn't mean they're unusable.
I wouldn't be offended, but like others said I would probably assume that you were religious. If I knew that you weren't religious, I would probably think it was a little odd, but if that doesn't bother you then go for it.
I agree with the poster above -- as an atheist, I believe the Bible is just another book (albeit one that's had undue influence over Western society). The reason Biblical names are in the mainstream name pool is because we live in a culture dominated by Christianity. I like the idea of secularizing these names as an act of rebellion! :P
Also -- I'm shocked to see so many people claiming they would assume a person is Christian if they used these names. Seriously? I actually know two Ezras, neither of them from Christian families. I know non-Christian Marys and Christian/Christinas. And my friend Abram is a secular Jew. Honestly, it seems like people tend to assume others share their beliefs for very little reason -- and that's their problem, not yours.
As Christianity's cultural aspects have been aggressively imposed on most of the world, I don't think you have to worry about a biblical name coming across as automatically religious, and in the same vein, I don't think any Christian gets to complain. It's not appropriation if it's been forced into your culture for centuries. Malachi might get some eyebrows. A name like Levi, some people might assume you're religious. But Ezra, I would think Jewish (and not necessarily religious) first. Other more mainstream biblical names (Abram, Asher), I don't think anyone would even bat an eye. I just wouldn't worry about it. If you love it, that's all that matters.