Many of her supporters are stating that her previous marriages were from before she found God and therefore cannot be used as an argument.
And all you do then is quote Matthew 5 to them. Specifically:
31 It hath been said, Whosoever shall put away his wife, let him give her a writing of divorcement:
32 But I say unto you, That whosoever shall put away his wife, saving for the cause of fornication, causeth her to commit adultery: and whosoever shall marry her that is divorced committeth adultery.
Oh, and Oniya has it right about the whole taking an oath... could always quote:
33 Again, ye have heard that it hath been said by them of old time, Thou shalt not forswear thyself, but shalt perform unto the Lord thine oaths
All in the book of Matthew, chapter 5.
And anyone claiming that she is born again now and nothing prior to her being saved counts - Jesus flat out stated his coming to earth did not invalidate the laws of the Old Testament.
17 Think not that I am come to destroy the law, or the prophets: I am not come to destroy, but to fulfil.
18 For verily I say unto you, Till heaven and earth pass, one jot or one tittle shall in no wise pass from the law, till all be fulfilled.
And this - http://www.huffingtonpost.com/zack-hunt/the-biblical-problem-with-kentucky-clerk-kim-davis_b_8071506.html
- is definitely worth a read.