The board in all likelihood can’t refuse her resignation.  This is a matter of state law, but most, if not all, state nonprofit corporation laws specifically provide that officers or directions may resign their positions.  The Model Nonprofit Corporation Law, which represents some of the best thinking on nonprofit governance, provides that an officer’s resignation is effective when notice is delivered unless some later time is specified in the notice.  Therefore, once she delivers the notice, she can leave her position confident that she has no more responsibilities as president.

It is a general principle of law that you can’t force a person to perform services.  Even if the person is under contract, the remedy is normally to sue for damages or to prevent the person from performing services elsewhere.  “You can’t force Caruso to sing” was the phrase when I went to law school.

Assuming that she is also on the board as a director, she will want to be sure that her resignation makes clear that she is also resigning as a director if she wants to leave entirely.  Otherwise, she may still be on the board, with all of the fiduciary responsibility that goes with the territory.