Memory-foam pillow: wash the cover, not the foam insert
For TEMPUR-material pillows, the removable cover can be washed as labeled, but the foam insert must not be washed or soaked.
Some pillows must not be machine-washed as a complete item. Tempur-Pedic says its TEMPUR material is sensitive to liquids: do not wash the foam, but wash a removable cover with mild detergent in cold water and air-dry. Other types of foam and fiber fill can have different rules; always inspect your own care label.