All reasoning requires some unprovable axioms. That's Godel's incompleteness theorem. A religious person can simply take the divinity of jesus (or whatever) as an axiom and use reason from that point onward.
But a non-believer can attack that axiom through historical record or the definition of divinity and point out how Jesus fell short (prophecies, etc). So if the axiom is falsifiable it is useless.
No, that's just reasoning. Godel's theorem is about the limitations of what you can build upon those axioms.
I'm not sure those limitations even matter much. What use is being able to prove "this statement is unprovable"?