I knew a couple Mennonite families that claimed they did not vote, because it was up to God to appoint a country's leaders. I don't know if that was the official teaching of their church, though.
I not a fan of those who use religion as an excuse to opt out of voting and otherwise being a good citizen. They enjoy the fruits of the tree but refuse to water it. Lazy bums if you ask me. Did you? Ask? Anyway I think every American citizen should pull his/her own weight. That includes participating and voting. Thank you for your reply Walt! :)
There are probably communities within various religions who avoid politics but it's not a requirement of most religions generally.
Hard-line Islamicists deny any need for elections because to them the only law is that of God, so governments exist simply to administer that. Ironically though the original caliphs were elected - though in what still amounted to one-"party" states.