Washing dishes or flatware does not, in of itself, violate any of the Shabbat rules. Nevertheless, there are some other considerations involved which certainly complicate the process. Here are the most basic ones:
a. It is forbidden to turn on the hot water tap on Shabbat. The reason for this is that when you turn on the hot water, you are releasing new water into the hot water tank that supplies the sink. This would be a violation of the prohibition against cooking on Shabbat.
b. It is forbidden to use a sponge or washcloth, this because using these materials invariably involves squeezing them to the point where water is expelled—a violation of the prohibition against "pressing (out liquid)" on Shabbat. Non-absorbent bristled brushes may be used instead.
c. Washing dishes is permitted on Shabbat provided that you may possibly need to use these dishes again on Shabbat. However, it is not permitted to wash dishes simply to spare yourself the hassle of doing so after Shabbat.
The last two rules also apply to major holidays. Using hot water, on the other hand, is permissible on holidays that do not fall on Shabbat, when cooking is permitted. (This applies to most water-heating systems, those whose pilot flame stays on all the time. One would not be allowed to turn on the hot water if the sink is hooked up to a system whose pilot ignites when the hot water tap is turned on.)
And one more note: It is permitted to ask non-Jewish household help to wash the dishes on Shabbat. If they choose to use hot water and/or a sponge, that is their prerogative. Since the act of washing dishes itself is permitted, the methods they choose to employ is not our concern.
