Each instance of dirt appearing in dreams is different but when it appears on your dishes it means that each dish represents a bad deed that you did and the fact that you are trying to put them into a dishwasher or to wash then should clue you in to finding out what this kind of dream means.
If you have a dream in which you see a dishwasher then this means that you need to clean up your life. There are a lot of different facets and aspects of your life that fit various places and serve a multitude of purposes but if you have a dream that you are using a dishwasher or that you are a dishwasher then this means that you need to work out your issues with yourself and make sure that you get yourself cleaned up. Each of the dishes that is dirty represents something wrong that you've done and the more they pile up in the sink the dirtier you feel inside, you better do something to clean them up before you go crazy inside.
If you have a dream in which you are a dishwasher and you are washing all kinds of dishes, not just your own, it means that you feel like you have to take on the world problems and you need to make up for all the wrongs committed by others. This is not the truth. In reality you are your own master and should be the controller of your own destiny. You have no obligation whatsoever to solve everyone else's problems and if this starts to become a burden on you, you should stop immediately.