The Shulhan Arukh 323:6 teaches that one is not permitted to exert oneself on Shabbat for something which will only be used after Shabbat. Therefore, one may not wash dishes which will not be used until after Shabbat. However, since drinking glasses may be used throughout the day, they may be washed on Shabbat.
There are, however, a few cases worthy of discussion.
First, R. Ovadia Yosef (Hazon Ovadia Helek 6 pages 100-104) writes that after making Kiddush on Shabbat morning, since the cup will not be use again on Shabbat, it should not be washed until after Shabbat. He adds, however, that one may simply rinse the cup with water, as that is not a tirha. He adds that one may also fill up a dirty pot with water in order that it will be easier to clean after Shabbat, as that is not considered to be a tirha.
Second, R. Ovadia Yosef (ibid.) also discusses whether one may clean his dishes on Friday night in order to use them the next day, if he has other clean dishes. He cites the Mishne Halachot who rules that one may wash those same dishes, and does not need to use the clean dishes in his closet.
Third, what if a person wishes to wash dishes for other reasons? For example, what if a person wishes to wash dishes because dirty dishes may attract bugs, or for hygienic reasons, or because guests may come and it is not proper to have dirty dishes in the sink? In this case R. Ovadia Yosef (ibid.) rules one may wash the dishes. Similarly, one may make a bed in the morning, even if he will not sleep until after Shabbat, as it makes the room look nicer.
Summary: It is prohibited to exert effort, i.e., a tirha, on Shabbat for something which will be used after Shabbat. However, the action does not involve a tirha, or if there is another reason to exert the effort, then it is permitted.