Is washing dishes on shabbos a isur deoiraiso or derabonon? The nafka mina would be whether one must protest in a case when his protest won’t be listened to.

Shalom, The level of severity of washing dishes on Shabbat with a sponge or wash-cloth depends on the manufacture of the sponge. This can depend on what it is made of, and whether it has a handle etc. There is also an argument amongst the Rishonim about this issue. But in answer to your question, my advice is not to protest in any event when you know that your protests are in vain. Rather, offer to do the dishes yourself (in an acceptable fashion). Blessings, D. Sperling.