The Catholic Church and other religious organizations are
private organizations, like other charitable organizations, so they do NOT have to open their books even if they are tax-exempt. Other private organizstions do not have to open their books to the public either (just to the BIR or to investors).
The separation of Church and state can only be effective if the government is not allowed to tax the Church. Taxation is a form of control and historically this has been used by the state to try to control religion. It is important to remember, however, that the separation of Church and state is designed to keep the government out of religious affairs (and
NOT keep religion out of politics since politics has a moral dimension). That is why religious organizations are tax-exempt: to preserve separation of Church and state.