Depends on the "dish soap". I've seen everything from sodium fatty acetates (old fashioned lye-and-grease soap, Castille soap) to PEGylated alkanes and sodium lauryl sulfate.

Vinegar and lemon juice work well on Lewis bases (amines, some proteins) and some inorganic salts, but they aren't detergents.

NaOCl and NaOH are right proper cleaning agents and will saponify fats.