There are several ways to remove gum from a shoe. One is just try to scrape it off or physically remove it by hand. Another way is to peanut butter. Peanut butter helps loosen the gum. Finally just good old soap and water has an affect as well.
To get gum off the rubber of your shoe, take an ice cube and rub it over the gum until it becomes hard. Then, when it's firm, it should remove easily. If this doesn't work, you will need to use a stain remover for carpets.