Update: September 2
Although she could face a contempt citation and personal fines, she cannot be "fired". That requires impeachment and the involvement of the state legislature, as NBC explains here. The Washington Post has a detailed biography of Kim Davis today here (a "Heaven and Hell" issue). Vox (German Lopez) explains further, and notes that for right now Davis has refused to issue all marriage licenses. Jonathan Adler of the Volokh Conspiracy explains a piece by Antonin Scalia which would show that Davis should resign if she cannot follow the law, here.
Update: September 3
Kim Davis has been held in contempt of court and is in custody of US Marshalls, In her absence, her office can issue marriage licenses (ThinkProgress story). Curiously, Donald Trump sidestepped a question about this Thursday afternoon.
Update: September 4
There are reports that she had been offered a chance to "stay out of jail" if she would allow her subordinates to offer the licenses, and she refused. The Post reports that the county is now offering marriage licenses again under federal court supervision, here.
Her attorney explained to CNN that her name would be on the license, and that is one of the problems. To me, that says she should resign from the job immediately. Jail confinement under a contempt citation is ironically a "civil action" under the court.