It refers back to "evidence":
Tell a devout Christian that his wife is cheating on him, or that frozen yogurt can make a man invisible, and he is likely to require as much evidence as anyone else, and to be persuaded only to the extent that you give it.
But there's also an omitted argument here, "to him":
Tell a devout Christian that his wife is cheating on him, or that frozen yogurt can make a man invisible, and he is likely to require as much evidence as anyone else, and to be persuaded only to the extent that you give that evidence to him.
It means that you'll need to present a devout Christian with evidence if you want to persuade him about matters unrelated to God.
The next sentence presents a contrast:
Tell him that the book he keeps by his bed was written by an invisible deity who will punish him with fire for eternity if he fails to accept its every incredible claim about the universe, and he seems to require no evidence whatsoever.
This says that he takes the existence of God on faith. In other words, he believes in God even without evidence that God exists. The paragraph as a whole presents this contrast in a negative light; the author is portraying the Christian faith as irrational.
Disclaimer: I'm not trying to express my own opinion here. I'm only trying to explain what the paragraph says.