The shed will be fine, but if there are any windows, make sure they are not in direct sunlight - that will degrade the rubber. I wouldn't put anything on them, but as puddinboo says, a good wash wouldn't hurt. Mine are stored indoors, up on a shelf, out of sunlight and sometimes, if the temperature is nice when I swap them, they get washed.
If you have r-compound or slick race tires, that's a different story. They need to be kept away from freezing temps. I have a pallet full of such tires I store by the huge boiler at my in-laws place. I call it "the tire spa".