It will vary by the individual costumes and their fabrication.
Zentai-Zentai.com which offers thousands of superhero suits say that morphsuits (made of Spandex Lycra, Silk Lycra and Velours) can be either hand-washed or just be thrown in the washing machine with the temperature set to cold while Shiny Metalic and PVC suits just need to be wiped clean with wet cloth.
A simple, single-color morphsuit is the easiest kind to deal with. With the use of a gentle cycle, they can occasionally be machine washed, but always refer to the packaging for wash care instructions.
Get more tips on how to wash a morphsuit, shared by eBay.com.
Check this out! BuyCostumes.com shares The 10 Best Things to do with a Morphsuit.