My religion demands that I neither pay nor receive interest. But if I pay my taxes late, then you better believe that I'll be paying interest. Being compelled to pay interest under threat of jail is a consequence of not paying taxes on time - therefore paying taxes on time is a religious duty.
First of all, neither the Qur'an nor the authentic hadith of the Prophet SAW demand that the face be fully covered. If this woman was part of some other religion that had scriptural requirements that supported full facial covering, then she might actually elicit some sympathy from me. However, her invocation of Islam here is disingenious on her part, since she demonstrates her intent to violate her own stated beliefs by choice.
Let's take her at her word. Suppose that her religion (whatever it may be) truly demands that she wear a full facial covering. The requirement by the State of Florida that her photo show her face is a requirement for the privelege to drive (neither the Constitution nor the Bill of Rights demand that driving is a right). Then by the constraints of the belief system she chooses to adopt, not driving becomes a religious duty
The bottom line is that this woman chooses to wear her full facial veil based on a personal interpretation of religion . She is attempting to leverage that personal interpretation to evade a civic requirement in order to lay claim a privelege. This puts he in the position of placing her desire to drive above her belief in her religious responsibilities.
She has to decide which is more important. If she was truly as committed to her personal interpretation as she claims from her high horse, then she should consider driving haram, just as I am personally forced to consider paying taxes late haram.
Note however that this issue is fundamentally different from the restrictions on Muslim schoolgirls in France from wearing the hijab. That is a true case of discriminating victimization by the State and interference with religion.
UPDATE: Al-Muhabajah makes some important relevant points and argues that even if this woman is wrong to demand a photo license with veil, the state is equally wrong to demand that her license be revoked outright.
UPDATE 2: Matthew distills the issue:
As I understood it, the state never denied that she has a right to wear a veil while driving. What they denied was that she has a right to wear a veil while getting her photo taken. If the state is really trying to prevent her from driving-while-veiled then I think she ought to win. If all they want is an unveiled photo and then want to let her wear what she wants, then I think the state should win.
 The irony of this compared to the opposite situation regarding women drivers in Saudi Arabia is not lost on me. and is grist for the comments mill.
 Which has essentially zero scriptural or doctrinal support. It's essentially a religious innovation (bida).