Some theologians like to say that God does not exist. They say this to mark the radical difference between God and creatures.
1. If one is going to say such things, a more helpful way to speak would be: "God exists but we don't." For that would still get across the radical difference between God and creatures, but get right the fact that God is the one who is the more real. Compared to God's reality, we are but shadows. It is said that God said to St Catherine of Siena: "I am who I am, and you are she who is not." This poetically conveys a deep truth. We are but shadows, and "shadow" is often an overstatement.
2. There are many metaphysicians who like to say that complex artifacts like tables, chairs and blowguns don't exist. But many of them say this only in philosophical contexts and not in "ordinary" contexts, or they qualify the "don't exist" with a "really". They may or may not be misguided in the form of their odd denial, but what they (we!) are getting at is plausible: There is a deep difference between the kind of being that a table, chair or blowgun has, and the kind of being that a horse or a photon have (some of these philosophers will class the horse with the chair; that's mistaken, but the basic point I am making isn't affected). The ordinary language sentences "The pig exists" and "The car exists" have very different (nonpropositional) grounds: the former is grounded in a single thing while the latter is grounded in the arrangement of many things. Well, these theologians, like these metaphysicians, are also impressed by a deep ontological difference (a deeper one, perhaps). But like the metaphysician who is willing to speak with nonphilosophers in ordinary ways, these theologians should be willing to say "God exists" in contexts of ordinary worship. Or like the metaphysician who says that computers don't really exist, she could simply make a qualification: "God doesn't exist in the shadowy way." Or, more perspicuously?, she could say: "We don't really exist, but God does." (Though I think that if one does that, one should also distinguish us from artifacts. Perhaps the distinction could be marked with "really" and "really really"!)