The priest pays income taxes and sales taxes. The church doesn't have to and the Constitution would probably need to be amended to change that. I say probably because there isn't a case directly on point. The closest you'll find is Walz v. Tax Comm'n of the City of New York, 397 U.S. 664 (1970).

A New Yorker challenged tax exemption that the state had granted to churches, essentially claiming that the exemption was a subsidy and establishment of religion. SCOTUS disagreed.

Some good nuggets of dicta

We find it unnecessary to justify the tax exemption on the social welfare services or "good works" that some churches perform for parishioners and others [...] To give emphasis to so variable an aspect of the work of religious bodies would introduce an element of governmental evaluation and standards as to the worth of particular social welfare programs, thus producing a kind of continuing day-to-day relationship which the policy of neutrality seeks to minimize. Hence, the use of a social welfare yardstick as a significant element to qualify for tax exemption could conceivably give rise to confrontations that could escalate to constitutional dimensions.

The exemption creates only a minimal and remote involvement between church and state, and far less than taxation of churches. It restricts the fiscal relationship between church and state, and tends to complement and reinforce the desired separation insulating each from the other.

All of the 50 States provide for tax exemption of places of worship, most of them doing so by constitutional guarantees. [...] Few concepts are more deeply embedded in the fabric of our national life, beginning with pre-Revolutionary colonial times, than for the government to exercise at the very least this kind of benevolent neutrality toward churches and religious exercise generally

It is obviously correct that no one acquires a vested or protected right in violation of the Constitution by long use, even when that span of time covers our entire national existence, and indeed predates it. Yet an unbroken practice of according the exemption to churches, openly and by affirmative state action, not covertly or by state inaction, is not something to be lightly cast aside.