Very rarely, JECE, and it's usually very hard to do. The Church of Scientology had a very long, very ugly battle with the IRS that it finally won, and it was a relatively obvious case. Also, there are pastors who break the political endorsement rules, and little is usually done about it.
Anyway, I'm opposed as well. I'm actually a little surprised that's even allowed under Alaska state law, but that's the beauty of having fifty different states, I guess.
Taxing nonprofits is one of the most direct ways to make life worse for society's most vulnerable; in many cases, that includes churches. (Some churches, it might make life directly worse for other sectors of society).
JECE, under typical definitions of non-profits, churches are clearly non-profits. There are no stockholders receiving dividends from left-over intake. It's true that in some large churches (as in some other non-profits), a scandalous amount of money is spent on salaries and on frivolous expenditures. That's far from universal, and I'd be surprised if Nome has any megachurches.