The phrase is often used metaphorically, meaning the person won't be hurt at all. Despite the heated argument, he insisted that he wouldn't harm a hair on her head.
Pro Tip 2/3
Emotional Assurance
Used to reassure someone that they will be completely safe. Don't worry, I'll make sure not a hair on your head is touched during the trip.
Pro Tip 3/3
Protective Statement
Can indicate a strong protective feeling the speaker has for someone else. As your older brother, I won't let anyone harm a hair on your head.