I have begun dating someone whom I have recently found out may be distantly related. I want to know what the exact relationship is. His maternal great-aunt married my maternal great-grandfather's 1st cousin. What is the relationship?

Sounds more like distant cousins --third cousins or second cousins once removed or something like that
Not your first cousin
Most states laws don't go much further than first cousins so you probably are okay.
What state[/country] do you live in?

There is absolutely no relationship between the two of you. You would have to have common grandparents, great-grandparents, great-great grandparents, etc to be related. The relationship between your two families is only by marriage and as such there is no relation. Just to be sure, I input your information into my genealogy program and it too, said there was no relation when I entered you to and used the relationship calculator.