One is not permitted to turn on the hot water tap when washing dishes on Shabbat since by doing so, additional cold water will flow into the hot water tank and be boiled on Shabbat. However, if one has prepared hot water before Shabbat, or a tank which stores the hot water, it may be used to wash the dishes. However, it is forbidden to pour hot water on something which is very greasy, since the hot water will dissolve the fat on Shabbat.
One may not use a bar of soap on Shabbat, but using liquid soap is permitted; however, some people are particular to dilute it, making it more liquid than usual, reducing its viscosity. One is not allowed to use a sponge or a steel wool on Shabbat, nor wash dishes that are not needed for use on Shabbat. If one wishes to soak the dishes when clearing them off from the table, as long as the basin of warm water was prepared before Shabbat, it is permissible.
Rav Yosef Yeshaya Braun, shlita, member of Beit Din of Crown Heights; One Minute Halacha, #137, reprinted from crownheightsconnect.com
