I live in Georgia and I'm 16. My dad has custody of me. I want to move in with my mom when I turn 17, but my dad says no. if I do move in with her can the law make me come back or can my mom get in trouble for letting move in?

This is a tough issue in GA actually a 17 can move out and there is nothing that the authorities can or will do,
** know this personally from my 17 year old moving out, yes that is you Jon I am talking about,

But your dad has a court order giving him custody over you and if you move in with your mom while he may not be abe to do anything to you, he can file against your mom for interfering with custody and even contempt of court.

** this is assuming your mom also lives in GA, since at 17 you have really no right to move outside of the state.

But at 17 even in GA, you still can't sign any contracts, rent or get electric.