There’s a lot of stuff on postmodern theology, but I think most of it is written as a reinterpretation of religion through a postmodern lens, or alternatively, a religious perspective of postmodernism.
It seems harder to find a skeptical analysis of the way that postmodern philosophy has affected religions and people’s propensity to have faith.
The closest thing I’ve been able to find is this:
Augustine and Postmodern Thought: A New Alliance against Modernity? (Bibliotheca Ephemeridum Theologicarum Lovaniensium)
This is an interesting book by a preacher who uses post-modernism as a means of luring in nonbelievers.
Preaching After God: Derrida, Caputo, and the Language of Postmodern Homiletics
There is also
Postmodernism, Reason and Religion
But that doesn’t seem to go into the effects in any depth.