You shouldn't have any damage resulting from cones, but you will have a lot of rubber and tar marks on the front/sides of you car. The cones get covered with rubber and tar from the hot tires and that gets spread all over your car when you inevitably hit them.
It usually comes off when I wash the car, but you may need to use clay in certain spots. You should probably wax/zaino the front of your car more often if you AutoX. It helps in getting the rubber and tar marks off.