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That would be 18 years of age for emancipation from ones parent in North Carolina this would include moving out. He must be 18 to legally move from his parents property unless he gets emancipated by a court of law which isn't easy to do. He would need to petition the courts for emancipation and if his parents do not consent it is unlikely this would be approved.
You are still legally responsible for him until he is 18. If he leaves and you fail to report him as a run away you can be held liable for any actions and for any incidents he has. Should he become hurt while away you could be charged with neglect as well.
Here is a link with further information on the subject:
Emancipation of Minors - All US States
I am sorry, it sounds like you have been misinformed. The parent can give the child permission to live somewhere else but the parent is still liable for the actions of the child. The state can also still step in and file neglect charges on the parent if they feel the child is doing without while living away from home.
Often these little stories circulate from different stories such as this. There is no law that allows a child to leave his parents home without permission in the state of North Carolina.