Yes and no actually. During the day, the sun is shining, and the black draws heat to the roof while white reflects it. During the night, the air cools down and the sun no longer shines, making the roof the heat source. So the black would cool faster, while white reflects heat back to the roof. But since the problem usually is worse in the day, you want a light/reflective surface to keep it as cool as possible.
There are insulated sheeting boards that work wonders inside the coop under the tin roof. Our contractor used it when they raised the attic during the remodel of our house. We seldom have to use the A/C because the house stays so comfortable and we have no shade trees around the house. Tin draws heat no matter what color it is so the extra insulated sheeting under the roof inside the coop will be a big help. The chickens can't peck at it either unless your perch is situated that they can reach the ceiling to peck at it.