Thought I'd do a run through without lifting a finger for the church.
You can't do absolutely nothing, or the main quest won't progress; you have to go to San Matheus and see the cardinal to ask her about the Malichor, you have to pass that loathsome inquisitor burning a guardian, and you can't avoid having Petrus foisted upon you. When you talk to the cardinal, you also can't avoid getting the Demoniacal Cult quest.
For the Cult quest you have to deal with Ephesia a couple of times, but that's all.
You don't need to go to Eden and do the Origins of Theleme quests at all. This does mean that you can't get all the campsites for de Courcillon, as you never get access to the sealed-off area in Veldivie, which contains two campsites.
You also have to deal with that loathsome inquisitor again to get the Nauts out of jail, but you can deal with Eseld to take out the camp of the Ordo Luminis, which annoys the cardinal so that's fine.
You need either to leave Kurt's quest unfinished so that you can leave San Matheus swinging for Torsten, or you can just not send the messenger.
When you're collecting allies for the endgame, you have to see the cardinal in her little house in New Serene. She doesn't seem too ticked off, though she says she has no soldiers to give you. She does say she'll give you a piece of helpful jewelry, but doesn't actually give you anything. Similarly, commander Torsten in San Matheus says he's giving you a weapon, but doesn't. Neither of them will join your alliance.
Petrus' quest hangs on in your journal until you start "Assault on the Heart".
Just for a change, I became a god at the end!