It sounds like you might need to do a bit more than just drying your brothers tennis shoes. If the shoes are covered in mud, you will need to first get the mud off and remove the stain and then dry them. Hopefully you didn't wait too long to try and clean them off because shoes caked in mud are going to be a lot harder to clean after they are dry. Depending on what color they are, and what they're made of, you can safely use a variety of cleaning agents to help you. Soap and water (use laundry detergent for the soap) is a good place to start, and then try some bleach if that is not doing the trick, and if the shoes are originally a white or lighter colored shoe. There are also some excellent home made shoe soaps that you can find recipes for, if you prefer to mix up your own.
For the most part, putting tennis shoes into the dryer is a BIG no-no. You have seen what the dryer can do to your favorite pair of shorts or shirt. The high heat causes shrinkage of cotton based products, and your shoes are no exception. The fabric parts of tennis shoes are typically made with at least a cotton blend, so you risk shrinking the shoes. High heat also is not good for the rubber soles, and you risk heating the rubber to a point that it gets too malleable and changes shape. The best way to dry shoes is by simply sticking them in the sun. You can also speed up the process by stuffing the shoes with newspaper. This works especially well if they are drenched through and the insides of the shoe are wet too. But you'll need to switch out the newspaper with new, dry newspaper at regular intervals because, while the newspaper is like a magnet for water and absorbs H2O quickly, it also becomes saturated quickly and once it is saturated through, it simply will not hold any more water. (If you absolutely must use a dryer to dry your shoes, then dry them by themselves with no other clothing on low heat on the tumble dry setting.)