Experimental studies have been done on intercessory prayer. The results are tantamount to prayer being a failed hypothesis.
If we expect certain gods to answer prayers, these experiments are clear evidence against their existence. If we expect the God of classical monotheism to answer prayers-as theists insist-then failed prayer is strong evidence against the existence of God (By the way, insisting that prayer isn’t merely about asking God for help in time of need is a red-herring).
Also, if God doesn’t answer prayers, she’s not worthy of our time or worship. Given all the horrific suffering in the world, the least God could do is answer prayers. Not doing so means that “God” is probably a dick or weak. For all we know, God has a mysterious good reason not to answer prayers. But for all we know, God also has a mysterious good reason to answer prayers. Thus, appealing to unknown reasons changes nothing; it also explains nothing.
One could say that God used to answer prayers but not anymore. However, this position is ad hoc and raises a million questions. It also appears to be a position that is inconsistent with the nature of an all-just entity.
Secondly, one could object by saying that we can’t test the supernatural with science because science is concerned with the natural. My response is that if the supernatural interacts with the natural, as theism posits, then science can test the supernatural.
Finally, a person could appeal to the mystery of God. But I already dealt with that in this post. Also, bringing out the mystery card almost always becomes a case of special pleading. It’s also not clear how appealing to mystery could possibly help when we are talking about very general facts (and general concepts). The upshot is that if we make our concept of God too abstract and mysterious, then the concept is useless and not worth our time.