There is no such bird. Any bird with a properly functioning digestive system will release parts of food that it does not need in the form of waste (poop). The only thing that would cause a bird or any living creature not to do so is constipation, which again is an unhealthy condition.
Because, for birds to poop in their nests, well that's one stupid bird. Birds poop everywhere, they do not "target" cars, people, or statues, etc. It just seems that way because these things matter more to us than grass, parking lots, etc. Admittedly, some birds do target other birds.