This depends on the materials your koozie is made of. For example, soft-sided koozies (neoprene, foam) are usually hand-washed or spot-cleaned with mild detergent and warm water. Hard-sided koozies are usually washed like regular dishes.
If you have unique koozies with extra features, make sure to follow the manufacturer’s instructions about care and maintenance. And, regardless of the type and how you clean it, make sure that your koozie is 100% dry before storage to prevent mold, rot, or mildew.