I want to do a free daily podcast of the Bible. Would I need permission of the copyright holder of the translation I am reading or would this be considered fair use? Would I need to state the source or, conversely, would I be restricted from stating the source without authorization?
First of all, yes, the (purported) author of the Bible has much better recourse than the standard Cease & Desist letter -- for example, He could send a plague of locusts or kill your kin and kine -- and arguments about "the life of the author" would probably make things worse rather than better, but, no, He probably won't take any sort of action.
And yes, the translator for any particular version does hold the copyright but there are many, many translations whose copyrights have long since expired. The King James Version (certainly the most poetic translation) saw its copyright expire before George Washington was born. There's also the now-public-domain American Standard Version.
Finally, there are versions released under very flexible licenses. For example, you can record the New English Translation without paying a dime so long as you aren't making money off it yourself.
And the translation certainly does matter. For proof, look no further than Genesis 1:2. The (free!) King James Version reads:
And the earth was without form, and void; and darkness was upon the face of the deep. And the Spirit of God moved upon the face of the waters.
But the (not free!) New International Version is
Now the earth was formless and empty, darkness was over the surface of the deep, and the Spirit of God was hovering over the waters.
There are those who claim that Shakespeare and Marlowe helped out on the KJV. I don't know if I believe that, but you don't have be a poet to realize that "hovering over the waters" is not the image you want to start your Bible with. Makes the Lord of Hosts sound like He's in a helicopter...
There are several issues here.
Issue 1: Making a recording of any copyrighted written work is subject to the same restrictions as making your own printed copy. If you were to buy a novel that is still protected by copyright and make an audio book of it without permission from the copyright owner and sell copies of this recording, they could sue you for copyright violation and easily win.
Issue 2: In general, copyright is good for the life of the author plus 70 years. So the copyright on the original Greek text of the New Testament ran out about 1900 years ago. The copyright on Genesis ran out something like 3300 years ago.
Issue 3: Translations have their own copyright, separate and distinct from the original work. So, for example, the New International Version was published in 1978. I presume (without checking, I don't think the detail is important here) that it's considered a "work for hire" and so had a 95 year copyright, so that won't run out until circa 2073.
Issue 4: You can quote short excerpts from a copyrighted work without permission. This is called "fair use". Exactly how much you can quote is decided on a case by case basis considering a variety of factors.
Issue 5: Many Bible translations try to simplify number 4 for you by giving explicit permission to copy specified amounts of the text. For example I think the Hohlman says you can copy up to 250 verses as long as you don't copy an entire book. (Some of the short books, like some of the shorter epistles, are less than 250 verses.)
So your choices are:
Record the original Greek and Hebrew. I'm guessing that's not practical for you.
Use a translation whose copyright has expired, like King James. (I've read that the UK has given the King James a perpetual copyright, and I see that statement repeated in one of the comments on another answer. But in the US, the copyright on King James ran out before there was a US.)
Don't record the entire Bible. Just record snippets. I don't know what you're trying to accomplish so this may or may not be practical.
Get permission from the copyright owners. They might ask for money, especially if you intend to sell your recordings. Or they might say no. But you can ask.