I have an S-Corp. For all intents and purposes all of the corporate "profits" are passed on to me each year as dividends.
Generally speaking, for the types of corporations that are not exempt from paying taxes, they only pay it on profits that they hold onto. If they spend the money it is a business expense and therefore not a part of the profit, therefore no taxes on it.
All the tax exempt status does for the church is let it them carry some money over from one year to the next without getting taxed. Any money they pay in wages is taxed. And most churches aren't actually sitting on any real piles of cash with only a few notable examples.
I don't care about their corporation being tax free, but I think they should have to pay property tax.