In one sense these two lines refer to the butler, who, because he interacts frequently with the Capulets and their aristocratic friends, should have the manners of a gentleman. The expression, “with unwashed hands” means to be in a hurry, and the butler is feeling so pressed for time that he’s speaking to the servants without his normal politeness, and that lack of politeness seems to the servants a “foul thing.” 

Shakespeare also puns on the word "manners," which in Latin means things held in the hand. So the good manners can refer not just to the butler’s behavior, but also to the nice furniture and the serving trays being cleared by the servants.  Because they have unclean hands from clearing the dirty dishes, that’s also a “foul thing.”