The most obvious one is the "god of the gaps" problem now applied to intelligence. If you look at things like Aquinas' Summa Theologiae [1] you'll see that there are explicit assertions that things like rationality transcend corporeal nature. As such the more examples we have of AI behaving in rational ways, we will be forced to conclude that either rationality does not transcend nature (removing the last remaining gap where we can invoke a supernatural explanation) OR that we have created artificial souls.

[1] https://www.newadvent.org/summa/1078.htm

I think theologians will hang on for a while saying "what AI is doing isn't really rationality". But eventually the theologians are going to face a reckoning: what AI is doing will look so much like rationality that they will be required to answer the question "what specifically differentiates the two?" and then they will be stuck. Neuroscience does not understand how our brains produce rationality nor do our AI scientists understand what algorithms their neural nets are implementing post-training. Therefore, any specific claim the theologians make runs the risk of immediately getting invalidated by some scientific discovery on either side (theologians these days are generally smart enough to avoid putting themselves in that position.)

Peter Thiel famously went to the Vatican and started giving lectures about AI and the Antichrist. Pretty rich for a guy who built his billionaire fortune on an AI-powered spying company.