"Do you start to do the washing up?"
You can but "te" at the beginning. However that way the meaning would be a bit different. By putting "te" there, the emphasis is on the person of the doer. (By this I mean, that the asker knows that the dishwashing will definitely and surely be starting, but it is unclear for them who is going to do it.) Without "te" it is more like a request in the form of a question, so it means something like "Will you (please) start washing the dishes?" or "I'm going to have you wash the dishes."
I am writing all these without regard to the original English sentence (Do you start to do the washing up?), because I don't know how to decode that... I can't imagine any situation where one would say that.