Bringing Down Holy Books

A reader sent in this question and I thought it would be good to see what the community thought:

I have some Mormon acquaintances that I have had some very lengthy discussions with. The Book of Mormon makes claims that are demonstrably false based on tested empirical evidence (DNA lineage of Native Americans for example). If I were to go after the validity of the Book of Mormon, would these sorts of empirical issues be a good first target, or do they fall under the “science” realm that is too easily shoveled aside with the “just have faith” response? If so, could you perhaps clarify a little on what the best ways to go after a religious book might be? Are we looking for internal contradictions, philosophically problematic statements, wildly magical or unreal stories that are so outlandish we would never believe them today?

What do you think is the best way to convince someone their holy book… isn’t?