Washing dishes on Shabbos:
It is permitted to wash dishes on Shabbos, for the need of Shabbos, such as if there still remains one meal for which one will need to use the utensils for eating, in which case it is permitted to wash them. If however one will not be eating any more Shabbos meals, such as after Shalosh Seudos [or after the 2nd meal in those homes which do not eat Shalosh Seudos at home] then it is forbidden to wash any dishes or cutlery, other than cups. [If, however, one decides to eat more past the 2nd/3rd meal then he may wash the dishes for the meal.]
When on Shabbos may the dishes be washed: All dishes/cutlery needed to be used on Shabbos may be washed anytime on Shabbos, even much time prior to the meal. Thus, one may wash the Friday night dishes immediately after the meal on Friday night, and he does not need to wait until the morning.
Washing cups: It is permitted to wash cups throughout the entire day of Shabbos, even after the final meal, unless one is certain that he will no longer need the cup, in which case it is forbidden to wash it. [Even if one only needs to drink from one cup], he may wash 10 cups for the purpose of using one of the cups to drink in the event that he wants a drink.
How many dishes may one wash? So long as there still remains one meal which he will eat on Shabbos, one may wash as many number of dishes and cutlery as he wants, even if he only needs to use one of those dishes for the meal. [This applies even if the dishes or cutlery are all of the same set.]
One may wash as many dishes as he wishes if there is still one remaining meal left to be eaten on Shabbos. After the final meal, dishes may not be washed with exception to cups, unless one knows for certain he will not be needing the cups until after Shabbos, in which case even cups may not be washed.
Q&A On washing Dishes
May one wash dirty dishes even if he has more clean dishes available?
Some Poskim rule it is better not to wash the dishes if there are clean dishes available. Others Poskim, however, rule it is completely forbidden. Other Poskim rule it is even initially permitted to wash dirty dishes for the meal even if one has clean dishes available to use. Practically the custom is to be lenient.
May one wash the dishes after his last meal if they are tarnishing the cleanliness of the house?
Yes. One may do so according to all opinions, even if he does not need to use them to eat another meal on Shabbos.
May one wash dishes on Shabbos if he will only be using them the next Shabbos?
May one soak the dishes in water after his last Shabbos meal?
If one is doing so merely so the food does not stick to the dishes, then soaking it is allowed. If, however, food is already stuck on and one desires to soak it in order to remove the food, then doing so is forbidden. It is however permitted to place the dishes in the sink as normal, and then proceed to wash one’s hands over it.
May one wash his food pots on Shabbos?
No, unless one plans on using the pot on Shabbos for a certain usage.
May one enter water into his food pot in order to let the pot soak?
If there is food stuck to the bottom of the pot, doing so is forbidden, as explained above.
May one wash his Kiddush cup out after Kiddush of the day meal?
If one is particular to only use the cup for Kiddush and Havdalah then it may not be washed after the daytime Kiddush unless one plans on using it. If, however, one is not particular in this respect then it may be washed without throughout the day as is the law by other cups. In all cases one may rinse out the wine and then drink some water out from the cup and place it on the drying rack.
 Admur 323:6-7; Michaber 323:6; Shabbos 118a
 The reason: As by doing so one is preparing for after Shabbos, and it is forbidden for one to trouble himself on Shabbos for the sake of after Shabbos. [Admur ibid; Rashi Shabbos ibid; Raavad Hasagos on Shabbos 23:7; See also Admur 302:10]
 Kaf Hachaim 323:42; Ketzos Hashulchan 146:16
 Admur 323:6; Elya Raba 323:8; Olas Shabbos 323:11; Kaf Hachaim 323:39
 Such as immediately after the previous meal. [Admur ibid] Vetzaruch Iyun why doing so does not transgress the Borer restrictions? See Piskeiy Teshuvos 319:23
 Admur 323:6
 Admur ibid; Michaber ibid; Shabbos ibid
The reason: As there is no set time for drinking. [Admur ibid; Rambam Shabbos 23:7]
 Admur ibid; M”A 323:9; Elya Raba 323:9; Maharil Minhagim Shabbos 27; Tosefes Shabbos 323:10; Kaf Hachaim 323:41, 43
 Machatzis Hashekel on M”A 323:8; Kaf Hachaim 323:39
 Admur ibid; M”A 323:8; Elya Raba 323:8; Rokeiach 55; Tosefta 13; Tosefes Shabbos 323:10; M”B 323:26; Kaf Hachaim 323:40
 Admur 323:6 and other Poskim ibid regarding cups, and the same applies for all eating utensils, as rules M”A ibid regarding setting up ten beds if one may need to use one of them; Ketzos Hashulchan 146 footnote 30; Piskeiy Tehsuvos 323:8
The reason: The reason for this is because once the Sages allowed washing a dish for the meal, they no longer restricted how many dishes one may wash as every dish washed can possibly be used on Shabbos. Thus, this is allowed even if one is certain that he will not need to use all the dishes washed. [Ketzos Hashulchan 146 footnote 30 in name of Machatzis Hashekel and Peri Megadim; See Betzeil Hachochma 4:130; Mishneh Halachos 6:80; Michzeh Eliyah 1:62; Teshuvos Vehanhagos 1:223; Piskeiy Teshuvos 323 footnote 43]
 See P”M 323 A”A 8
 See Vayaan Yosef O.C. 167; Piskeiy Teshuvos 323:8
 Minchas Shabbos 80:254; Tosefes Shabbos 323:8; Nezer Yisrael Likkutei Rima 20:3; Aruch Hashulchan 323:7; Kaf Hachaim 323:39; Betzeil Hachachmah 4:130
 Beir Moshe 6:82 in name of Ohel Moed
 Bris Olam Ofah 90; Az Nidbaru 2:29; 4:36; Sheivet Halevy 5:39; 6:42; Mishneh Halachos 3:40, 41; 6:80; 11:287
 Poskim ibid; Piskeiy Teshuvos ibid
 See Beir Moshe ibid; SSH”K 12:2 ;Vayaan Yosef 2:295; Piskeiy Teshuvos 323:1
 This is similar to the allowance to make the beds on Shabbos morning even though they will not be used until the after Shabbos.
 Tehila Ledavid 302:6
 See Piskeiy Teshuvos 323:1
 SSH”K 12:3; Or Letziyon 2:34-1; Maor Hashabbos 1:3 footnote 29; Minchas Shlomo 2:35-12; Michzeh Eliyahu 1:56-2; Beir Sarim 1:44; Piskeiy Teshuvos 323 footnote 53
 Az Nidbaru 5:36
 As pots are not a meal utensil and are rather used for the cooking. Hence cleaning them serves no benefit for the meal.