Preparing for a Weekday on Shabbat
The Gemara in Masechet Shabbat (118a) teaches us that one may not prepare from Shabbat for weekdays, i.e. one may not perform an action on Shabbat if it is not necessary for the day of Shabbat itself. For instance, one may not warm up a food on Shabbat to be eaten on Motza’ei Shabbat.
The Gemara (ibid.) continues that it is likewise forbidden to wash dishes on Shabbat when this is being done for Motza’ei Shabbat, meaning that there is no longer use for them on Shabbat. The reason for this prohibition is so that one does not exert one’s self on Shabbat for a matter that is not necessary for the actual honor of Shabbat (Ra’avad Chapter 23, Halacha 7 of Hilchot Shabbat and Magid Mishneh ibid. in the name of the Rambam).
Washing Dishes after the Third Shabbat Meal
Maran Ha’Shulchan Aruch (Chapter 323) states: “One may wash dishes on Shabbat in honor of the day, for instance, if one still has a meal remaining. However, after the third Shabbat meal, dishes may not be washed. Vessels used for drinking may be washed all day, for one is able to drink the entire day.”
Based on this, vessels used for drinking, such as glasses and the like, may be washed even after the third Shabbat meal, since it is entirely possible that these vessels may be used after the meal. However, vessels used for eating, such as plates, silverware, and the like, may not be washed after the third Shabbat meal.
Washing Glasses after Sunset
Clearly, if the third Shabbat meal was eaten at a late hour such that it was concluded after sunset, glasses and other vessels used for drinking may not be washed after the meal since it is prohibited to drink after the meal (following sunset) until Havdala is performed.
Washing Dishes Which Are Necessary for Use after the Third Shabbat Meal
The opposite applies as well regarding dishes used for eating, such as plates and silverware, in that if the third Shabbat meal was eaten at an early hour and the family would like to use these dishes again, they may be washed even after the third Shabbat meal, for the only reason this is usually prohibited is because one does not generally eat anymore after the third Shabbat meal; however, if one knows that one will make use of these dishes again on Shabbat, they may be washed.
Although Maran zt”l rules stringently on this matter in his Livyat Chen (page 103) based on the opinion of the Meiri who prohibits washing dishes after the third Shabbat meal even if one intends to use the dishes again on Shabbat, nevertheless, he has since found a ruling in the Tosafot Rid (Rabbeinu Yeshaya of Tarani) that as long as one intends to use the dishes again on Shabbat, they may be washed even after the third Shabbat meal. (It seems that the Bet Yosef in Chapter 514 rules likewise.) Thus, one may be lenient in this regard.
Summary: One may wash dishes on Shabbat when these dishes are necessary for Shabbat itself. However, after eating the third Shabbat meal when one does not intend to use these dishes anymore, one may not wash them.
Glasses and other vessels used for drinking may be washed even after the third Shabbat meal (before sunset) since it is possible that they will still be used on Shabbat itself.
In the following Halacha, we shall, G-d-willing, deal with this matter further.