You can in fact prove the non existence of a thing that is logically incoherent. Obviously the default position to be is agnostic, but you can actually disprove the existence of specifically a tri omni God via the problem of evil.
If an all knowing, all powerful, all loving being existed, we would not observe evil in the world as it would be knowledgeable enough, powerful enough, and care enough to get rid of it. We observe evil, so this being does not exist.
Of course, a lot of behaviour of God in the bible suggests that he is not all loving, which would trivially resolve the paradox, but a lot of Christians believe in a tri omni being anyway, which makes my prior argument non entirely irrelevant.