My sister has been in an abusive relationship with the father of her daughter for the last two years. He is mostly emotionally and verbally abusive. We have all become increasingly worried about the safety of her and her daughter. She wants to leave... But he won't let her. Is it illegal for her to (secretly) leave the state until a custody hearing can be set up? Or does she need to file for sole custody first? Can we request that he submits to a psychological evaluation?

If she leaves the state with the child, she could be ordered to return the child to the state.
But she can leave, go to a shelter, get a job, get her own apartment,
As soon as he says she "can't" leave, she should

I suggest calling a local Family Law attorney who offers a free consultation to new clients.

In general its not illegal to leave the state with your own child. It becomes a legal issue when there already IS a custody arrangement that forbids that from happening. However, to protect your sister since custody will become an issue, I recommend meeting someone who can give you professional advise based on where she lives.

Good luck to you all ~ I hope she can get way for her sake as well as her daughter. Their safety is the #1 priority.