The best way to dry your shoes after a rainy run? Remove the insoles and then stuff the shoes with crumpled-up newspaper. Wrap more newspaper around the insoles. If it’s been an especially drenching run, you might have to restuff the shoes once or twice before they’re dry. Unfortunately, I can tell you from personal experience that getting shoes even close to a significant heat source isn’t the way to go—the glues and rubber and whatnot rapidly disintegrate and you’re left with an unsalvageable shoe.