No. Varys is entirely focused on the game of thrones. I don’t know if he’ll be alive long enough to hear that the Wall has come down and that the Army of the Dead is on the march, but that’s almost a different “Age of Man” from the one he understands and operates in.
I have to believe that Varys’ hatred for the supernatural will have to come into play at some point. That speech he gave to Tyrion about how he became a eunuch was too lengthy and too well-executed to just be a character beat without any kind of payoff.
Good point. Maybe he ignores a warning or a prophecy, or maybe he gets into some personal thing with Marwyn?