The best way to do it would probably be to get some venetians nobles, how were garrisoning the republic colonies in the balkans, to go natives. As soon you do that you just gain a foothold for greeks orthodixism in Venice. After that they can gain power and eventually takeover the governement. As for the effects I think it is everyone guess. I, for my part, see one big advantage in it for Venice: less restive colonies since they are not govern by catholitics italians, and one big disadvantage: the damages done by the inevitable power struggles that will be needed to truly dislodge catholicism as the official state religion.