Does cheap "plasticy" blue painter tape actually melt when printed upon?
It really depends on the tape. In my case, I've used various types of blue painters tape in bed adhesion situations and have only had one occasion where it sticks to the part. (All experience is with PLA) Best advice I can give is give it a shot, and be sure to wait for the part to cool before removing it from the bed for the best chance at not having tape stuck to the part.