Clerics of any religion do in fact not usually work on a fee for service basis, rather they are supported by more general income sources like tithes/income from church-owned property or investment/donations/state subsidies etc.
But are there exceptions? There are a significant number of fiercely and proudly pro-capitalist clergy, particularly in the US. Is there any church where, when a parishioner says “Pastor, I am deeply troubled. Could we have a talk?” the answer will be on the lines of “Of course, come along to my study. That will be $100/hour, billed in 15-minutes increments.”?
I think paying the clergy person directly would have negative tax consequences for him or her. I’ve never heard of paying for counselling from a clergyperson, but I could see it for other services like a wedding where the couple does not attend the church. Even then, though, the payment would typically go to the church (with the clergy perhaps receiving a tip or gratuity).
I’ve never heard of clergy charging a fee for pastoral care. I have experienced a minister refusing an honorarium for performing a wedding for non-members.
Charging to minister to church members would probably go over as well as a turd in the punchbowl – I can’t imagine anyone getting away with it in the churches I’m familiar with. As for non-members, the clergy at our church are so busy with their assigned duties I don’t see where they’d have the time to take on clients.
Fees for weddings and speaking engagements (for example, hiring another church’s pastor to do your church’s retreat) are pretty common.
A Presbyterian church I attended had a therapist on staff who billed by the hour (though the therapist used a sliding fee scale so that you paid only what you could afford; the church’s funds covered patients who couldn’t pay). But even there, you could still see the pastor about more general-purpose concerns without any fee. Every pastor I’ve ever known has drawn a line between pastoral counseling and therapy, and felt that both had their place.
As for tax implications - this kind of thing would need to classified property and generally reported as self-employment income, but the implications are not necessarily negative unless you count paying taxes as a negative thing in and of itself.