Question: Is it permissible to use disposable dishes and plasticware on Shabbat, or does it impinge on the honor of Shabbat?
Answer: It is permissible. It potentially saves time washing dishes on Shabbat in a permissible manner of course) for the next meal and also saves one from the stress of washing all of the dishes after Shabbat, and this itself brings "Oneg Shabbat – Joy of Shabbat". If one is able, it is preferable to use beautiful disposable dishes.
Ha-Rav Yosef Shalom Elyashiv was similarly asked: In a family blessed with many children, there are many dishes used during the course of Shabbat. In order to lighten the load of washing all of the dishes, the husband wanted to use disposable dishes, including a disposable tablecloth, so that after the meal they could simply roll up all of the dishes in the tablecloth and throw them in the garbage. The wife, however, asked: Even though it would certainly make things easier, isn't using disposable dishes disrespectful to the honor of Shabbat? After all, if an important guest came to one's house, wouldn't we bring out the fancy dishes? Rav Elyashiv responded: There is no impingement in using disposable dishes, and there is no disrespect to the honor of Shabbat (Va-Yishma Moshe Volume 1 p. 106).
Ha-Rav Yitzchak Zilberstein asked his brother-in-law, Ha-Rav Chaim Kanievsky this same question, and Rav Kanievsky answered with a story from his uncle, the Chazon Ish. As is known, the Yeshiva world honors Shabbat by wearing a nice tie. A Yeshiva student once approached the Chazon Ish and described how difficult it is for him to wear a tie in the summer because he sweats a lot. He therefore asked: Is it permissible not to wear a tie or is it disrespectful to the honor of the Shabbat? The Chazon Ish answered that if there is no enjoyment of Shabbat, there is no honor, i.e. if the Yeshiva student does not enjoy wearing the tie, than there is no honoring of Shabbat in doing so.
According to the Chazon Ish's answer, we can also say in our case, since washing the dishes can be a great stress, using disposable dishes is therefore not disrespectful to the honor of Shabbat. And Ha-Rav Yitzchak Zilberstein says that since there are fancy disposable utensils today, it is preferable to use them rather than the simple ones. Ha-Rav Chaim Kanievski agrees with him, although he holds that the basic Halachah is that it is even permissible to use simple disposable utensils and there is still no impingement on the honor of Shabbat.
Ha-Rav Zilberstein also added that lightening the burden on the wife/mother is in and of itself honoring Shabbat (Aleinu Le-Shabei'ach – Shemot p. 530).
There was once a young couple who was very close to the Bostoner Rebbe and Rebbetzin. The couple was also close to Ha-Rav Yosef Solovietchik, who was Rav in Boston, along with teaching at Yeshivat Rabbenu Yitzchak Elchanan. The couple was once invited to Rav Soloveitchik's home for a Shabbat meal. The Bostoner Rebbetzin asked the young woman: What did you see there? She answered: It was quite similar to what you do but there was one difference: They use disposable utensils. The reason is that Rav Soloveitchik's wife wants to participate in her husband's Motzaei Shabbat class, and if she needed to wash dishes, she wouldn't be able to do so. The Bostoner Rebbetzin went to her husband and told him this practice of Rav and Rebbetzin Solovietchik and asked: I am willing to eat on China every meal, but we have 30-40 guests every Shabbat and I wash dishes until Tuesday. Why can't I use disposable dishes? The Bostoner Rebbe said: You can use disposable dishes. The Bostoner Rebbetzin said that she is so grateful to this young woman who told her what she saw at the house of Rav and Rebbetzin Soloveitchik (The Bostoner Rebbetzin Remembers pp. 165-166).
Nonethess, Ha-Rav Shammai Kehat Ha-Cohain Gross, Rav of Kehilat Machzekei Ha-Dat of Belzer Chasidim and author of Shut Shevet Ha-Kehati, holds that since one would not use disposable utensils for an important guest or at a wedding, one should be strict not to use them on Shabbat for adults, but can be lenient with using them for children (Kuntres Dvar Hashem Zu Halachah – Tefilah U-Bar Mitzvah #6).
If someone is adamant that NON-disposable dishes should be used, he should roll up his sleeves and help his wife wash dishes after Shabbat. Satmar Chasidim end Shabbat very late due to a long Seudat Shelishit and the Rebbe's talk. Once, on a Saturday night, the Satmar Rebbe saw that one of his Chasidim was the last one in the Beit Midrash and was folding his Talit with great precision. The Rebbe asked him what he was doing. The Chasid said that he saw in various books that care in folding one's Talit is a Segulah for Shalom Bayit. The Rebbe responded: A better Segulah is to go home and help your wife wash dishes (Others tell this story in the name of Ha-Rav Chaim Shemuelitz. In the book: "U-Piro Matok – Bereshit" of Ha-Rav Yitzchak Zilberstein p. 140).