Who's burden of proof is it?

This example brings up a lot of issues but two pertinent fallacies in the book (Dr Bo’s) are: proving non-existence and shifting of the burden of proof 

I don’t think claims about the Bible can be proven logically.   People talk about having “faith”, which seems to be a way of saying—something may be true (and is true) even though it can’t be proven logically.   And if it can’t be proven logically then logical fallacies don’t apply.  

Whether this is the ultimate truth, or a way of winning by saying the rules (of logic) do not apply, everyone will have to make up their own minds.  

I think you made the wrong statement. Rephrase it to "I don't believe [insert issue] because I have not seen sufficient evidence of its validity." Then your discussion partner bears the burden of proof to convince you.

However, phrased the way you have it, you did make a claim and now you have to go through all the reasons you think the bible did not meet its burden of proof. The fallacy of trying to prove a negative. When you do not believe something and the other person does, I found it is a better discussion technique to ask the person where you missed the point and let them try to convince you. My experience is that if they are in error, they see the error of their ways or suffer cognitive dissonance. If they are correct, then you have learned something.

Your friend asked you to prove that the Bible has not met the burden of proof, right? Well the proof is this:
The Bible is a book. Anything that is in that book is a claim. There are no other evidence for what is written in the Bible about any of the people or incidents that it contains. That is the proof that it has not met the burden of proof.

If your friend question that argument tell him that believing that the Bible is adequate proof for the existence of god would result in taking Spiderman #1 as adequate proof for the existence of Spiderman. (that's reductio ad absurdum by the way)

So with that argument you have proven that Bible has not met the burden of proof. And now you should demand that THEY prove that the Bible has met the burden of proof.