If a restraining order is ordered on a father of a child by the mother, can the mother legally leave the state?

It depends on how the order reads.  If the order places a restriction on the mother as well and instructs her not to leave the state, then the mother would need to get permission from the court prior to moving.  If the order only restricts the father and places no restrictions on the mother, then she can leave the state.  The bottom line is:  follow the orders.  Most orders do not contain restrictions on the movement of one parent-- but they will often include notice requirements.  If mother plans to move and her order requires her to give father some type of notice, then the mother should comply with the notice provisions.

As a side note, if the mother has a pending custody case in one state, and then moves to another, she will still need to travel back to the original state to finalize the custody orders as many restraining orders are temporary.  If a mother is close to resolving the custody suit, it would probably be easier for her to remain in the state until the suit is finished.  In the end, however, she needs to do what is best for her and her child.