One of the less sympathetic features of the case, at least to me, is that her action is entirely symbolic. It does not prevent gay couples from getting married and it only prevented them from getting married in her town for a few days. That makes me suspect, perhaps unfairly, that she is acting not out of religious conviction but either because she likes attention or because she hopes to use the controversy to jump start a political career.
Which gets me wondering how people would feel about her action if it was more than symbolic, if by disobeying a legal rule she disapproves of she could actually keep it from taking effect. It is hard to imagine a version of her story that achieves that, but consider the same issue in a more plausible context:
You are a law enforcement official charged with enforcing the law against marijuana in a state with severe penalties for its violation. You believe marijuana should be legal. You can quietly subvert the application of the law by failing to follow up evidence of marijuana usage, recommending on spurious grounds against prosecution of arrested users, perhaps sending anonymous warnings to targets of investigations by other officers. You expect that you can get away with such actions for many years, since those supervising you are either sympathetic or incompetent. The result will be to save hundreds of people from arrest, conviction, and imprisonment.
Should you do it or should you resign?
For those who think it obvious that you should resign, that obedience to the law takes priority over moral beliefs, consider two real world situations along somewhat similar lines.
1. Jury nullification. If you are on the jury trying someone for a crime of which you believe him guilty but that you do not believe ought to be illegal, should you vote for conviction or acquittal?
2. President Obama's decision not to prosecute a specified subset of illegal immigrants.