Silk pillowcases can be washed either by hand or in the washing machine, provided you treat it gently enough. If you’re using a machine, avoid hot water and wash it on a “delicates” setting if possible. Do not use any bleach-based products, as bleach can be highly damaging to silk.
Unless marked otherwise, assume that silk products cannot be tumble dried. Instead, leave them out to dry in an area without a lot of direct sunlight — too much of the stuff can weaken your silk.