Answer: Rambam (Shabbos 23:7) writes that one is not allowed to wash dishes on Shabbos for use afterward. Such hachana, preparation, is prohibited as it is akin to mesaken, fixing things. The Ra’avad, however, explains that it is prohibited because it is an unnecessary tircha, effort.
R’ Yisroel Dovid Harfenes (Nishmas Shabbos 4:364) suggests that this machlokes would make a difference to our scenario. According to Rambam one wouldn’t be allowed to ask a non-Jewish person to do the melacha on their behalf while according to Raavad, it should be permitted as there is no reason to prevent non-Jewish people from expending effort on Shabbos. While the poskim generally follow Raavad in this machlokes, he writes that the poskim do not allow asking non-Jews to perform hachana unnecessarily.
The Magen Avraham (510:13) writes that not only is one forbidden from making cheese on Yom Tov because it is a tircha, but one mustn’t ask a non-Jewish person to do it on their behalf either (See ibid. 321:7).
The Elya Rabba (252:12) and Pri Megadim (Eshel Avraham 503) are more lenient, however, allowing one to ask a non-Jewish person to perform an act of hachana, providing that no other melacha is involved.
Nonetheless, the Shulchan Aruch Harav (OC 319:18) and Mishna Berura (254:43; 319:62; 510:23) allow one to be lenient in extenuating circumstances, however, such as to prevent significant financial loss (See Machazeh Eliyahu 63:35; Sharaga Hameir 2:42:7).
Thus, R’ Moshe Sternbuch (Teshuvos Vehanhagos 1:284) writes that if one employed staff to work for an hourly rate on Shabbos, one wouldn’t be able to instruct them to wash the dishes. If one stipulated that one needed it done, for example, by Sunday afternoon, then the waiters are free to wash the dishes whenever they want. Likewise, if one merely states that the dishes are dirty and they understand that they need cleaning, they are allowed to wash them.
In conclusion, one should not ask a non-Jewish person to wash dishes for them on Shabbos for later use, though one can tell them that the dishes are dirty and allow them to wash them.