Why is it permitted to use warm water on Shabbat “to wash dishes etc., but not to wash himself, even his face, hands and feet” ? As long at the temperature is below Yad Soledet Bo, why does it matter what the warm water is used for?
The rabbis' prohibition of washing on Shabbat with hot water, is even when not "Yad Soledet Bo" and only on washing one's body not on anything else such as dishes. The reason is because of “Gezeirat Habalanim” [Shabbat, 40] lest this will lead to people heating up water on Shabbat in a forbidden manner. (שו"ע סימן שכו סעיף א ובמ"ב ס"ק א, וסעיף ד ובמ"ב ס"ק יז, שש"כ פרק יד סעיף ב, ופרק יד הערה ג בשם התהלה לדוד). Perhaps the reason our rabbis differentiated between washing one's body and other objects is because when one washes himself, if the water is too cold, he is suffering and will be very desperate to heat up the water, and may come to do so in a forbidden manner on Shabbat. This reason obviously does not apply when washing an object such as dishes.