I could not work with a therapist who had opposing political ideology. While I have not asked my T directly, she has expressed agreement with my political views. As an atheist, immigrant, bisexual woman of color who is very interested in politics as a "hobby" and was involved in the Hillary primary campaign, I would not feel comfortable with a therapist who was very conservative/religious. I also think that what I have to say would make her cringe.
My parents live in the South, and my mental health/development has been deeply affected by people who identified as very religious/social conservative saying horribly prejudiced things to me. This being said, I know not all conservatives are prejudiced, and I know not all liberals are "open minded".
I guess, for me, it wouldn't be the politics per se, but rather her views on issues of gender equality, sexual orientation, immigration, and race...however, these things are closely tied to political orientation.
I have psychologist friends and they do find it very unpleasant when patients say racist/homophobic things (psychologists in general tend to be much more liberal/secular as a whole, than patients).
I actually think this is an important thing to ask during an intake interview/phone screen (at least for me).