I have washed my DDs dolls before in the machine, gentle cycle. Although only when desperate, hand washing is recommended IIRC. I didn't put them in the dryer, just air dried. We don't have much humidity here so I wasn't concerned with them getting musty. You could put them outside to dry if it concerns you.
I will add that even with hand and/or machine washing they still look pretty dingy. I think of them as well loved.