Although it is forbidden on Shabat to wash dirty dishes that will be needed after Shabat, one may wash dishes that are needed later during Shabat. For example, after Friday night dinner one may wash dishes and cutlery that will be needed for Shabat lunch, and after Shabat lunch one may wash dishes for Seudah Shelishit. As Seudah Shelishit is generally completed soon before the conclusion of Shabat, it would not be permissible to wash dishes after Seudah Shelishit, though if one does plan on eating again after Seudah Shelishit before the end of Shabat, then he may wash the dishes. Cups and glasses may be washed because people often need to drink and therefore, even after Seudah Shelishit it is permissible to wash drinking utensils. This is the ruling of Rav Ovadyah Yosef (a”h).
Rav Moshe Feinstein (a”h) writes that if one wants to put dirty dishes in the dishwasher after Seudah Shelishit for the sake of orderliness, so the sink will be clean, he may, as long as he places them in randomly and does not set them up to be washed after Shabat.
It is forbidden to use sponges on Shabat as this violates a Torah prohibition, as using them invariably entails extracting absorbed liquid. It is permissible, however, to use a plastic sponge that does not absorb water. It is also permissible to wear dishwashing gloves while washing dishes on Shabat, since they do not absorb water.