For cleaning manual coffee grinder burrs along with other small parts, obtain a bowl of soap and water and dip a Q Tip inside it. After that you can make use of the Q Tip to gently scrub the micro-grounds and oils from the smaller sized parts. I do not suggest while using dishwasher to clean any area of the grinder.
Similarly, how can you clean an espresso grinder?
Usually, the removable areas of the grinder are top-rack is it dishwasher safe (utilizing a gentle cycle), or wash them