The first thing that you should do after you take off your shoes is clean them. Caked on mud, stones caught in the tread, a pile you stepped in that was left behind by a dog; these should all be removed before you store your shoes. First, in the vein of some recent tips, your shoes will dry faster if there is not a layer of moisture and insulation between the fabric of your shoes and the air outside them.
Second, and more importantly, you will not have to worry about any chemicals in the mud (or other “dirt”) that could break down the fabric or the rubber in your shoes. You will prolong the life of your shoes if you minimize all the bacteria that would love to feed off of them.
Third, by removing the mud and any stones on your shoes, you will have an easier time of it. You will not have to waste your time before your next run cleaning them off, or else suffer the beginning of a run with very heavy and off balance shoes or with a stone clicking on the road and throwing off your stride.