Soap destroys germs by wedging into them and essentially pulling them apart. The coronavirus can be thought of as a grease ball. Soap is a bit like a magnet: One part of soap molecules is attracted to water and repelled by the grease (fats/proteins that surround the virus’ genome). The other part of soap molecules is attracted to fats but repelled by water. This is a lethal combination for anything soap comes into contact with that involves fats/proteins and water, including coronavirus.
The detergent we use to wash our clothes accomplishes essentially the same thing as soap. Wash your laundry as you normally do, but don't overload on detergent. This won't make a difference, and it can ruin your laundry machine. High heat wash and dry cycles are only recommended if the person wearing the clothes in question is actively sick (or came into contact with someone who was actively sick), and therefore shedding larger amounts of virus into their clothing (such as by coughing/sneezing directly into their shirt).
Drafted 17 March 2020