Answer: The Mishna (Shabbos 121a) teaches that one mustn’t ask a non-Jewish person to perform melacha on their behalf on Shabbos. While the Levush (OC 243:1) and Elya Rabba (243:1) write that this prohibition, amira leakum, is mideoraisa, the consensus of poskim (Beis Yosef OC 244; Mishna Berura 243:5) is that it is miderabanan (See Gemara Gittin 8b).
The Taz (OC 276:5) writes that one may instruct a non-Jewish person to wash dishes on Friday night even though they will have to switch the light on to do so. As they are doing so for themselves, that isn’t considered to be amira leakum (See Mishna Berura 276:27). R’ Yehoshua Neuwirth (Shemiras Shabbos Kehilchasa 30:23) explains that while one may ask them to do something that inevitably means they will be doing a melacha, that only applies if one isn’t going to benefit directly from this melacha. Thus, one may ask them to wash the dishes even though they will switch the hot water on, though one shouldn’t do so if they’re planning on washing up with them and using that water. If they choose to use the dishwasher, they may do so, too.
The Rema (OC 252:5) writes that one mustn’t have machinery operate on Shabbos if it creates a noise (avsha milsa). While some argue that one mustn’t have a dishwasher running in one’s house because of avsha milsa (See Shulchan Shlomo 252:14), R’ Moshe Feinstein (Igros Moshe OC 4:70:6) writes that one may set an alarm clock for Shabbos providing it can’t be heard outside of the room. As modern dishwashers are typically very quiet when they run, it would seem that even according to these poskim¸ they would pose no such problem.
In conclusion, one may ask a non-Jewish person to wash the dishes, even if one knows that they will most probably do so by performing a melacha, providing that they don’t ask them directly to perform it in that manner.