Yes, ironing works on paper, even thin Bible pages, provided you are careful and start gently.
Just remember that a well-crinkled Bible is often a sign of godliness. Smudges, curled pages, folds, and coffee stains are a witness to everyday usage. They show that the Bible does not stay safely on some shelf, but is out amid the muck of life where children and pets might encounter it. Besides, twenty years from now when that man's two-year-old has moved out of the house and isn't even coming home for Christmas, the man may turn to the opening chapters of Matthew and find that those crinkles bring precious nostalgia.