No. When I've run through mud, dog poop, and other "gross" things, I rinse my feet of those things before I head inside. A simple douse with the hose and the muck is gone.
On a normal day I can walk right inside. The bottoms of the feet do not collect dirt like the tread of a sneaker. If anything does get attached, the footsteps quickly rub it away.
Of course if I never bathed my feet would get filthy, just like the rest of me.