If I have legal custody of my nephew and I let him go live with mother out of state, is it against the law?

If you have legal custody per a court order of a child and you have the exclusive right to determine where the child will live, then you can allow the child to live where ever you chose.  There are a couple of exceptions, however. The first is if your court order limits your power.  For example, if the order requires you to also get an agreement from another conservator or guardian, then you would need to comply with any specialized provisions.  The second is if the court order limits the mother's access.  If the court order states that the child is to only have supervised or limited access to the mother because she was previously deemed unfit, then you could not let the child go live with her.  

Outside of the court orders, if you know that the mother is not fit to care for the child and you allow the child to go into a dangerous or potentially harmful situation, then you could potentially be charged with child endangerment.  If the child is harmed while in her care and criminal charges are not filed, you could still lose custody for not providing adequate care for the child.  You really need to be sure that where the child is going to live is a safe and healthy environment, because as the conservator of the child, that is your responsibility.  If mom lost custody mainly because of her age or finances and is now financially stable-- then this is a little easier to handle.  If mom lost custody because she was abusive or had drug issues, you might want to start out with visitations to make sure that the homelife is good before you allow the child to just move away, especially to another state where it will be harder for you to access the child on a regular basis.