A dishwasher may not be activated on Shabbat. This is because one may not use electricity, and additionally because the dishwasher heats water to wash the dishes. This heating is a transgression of Bishul.
One who always clears off dirty dishes from the table and places them directly in the dishwasher may do so on Shabbat as well. Then after Shabbat he can run the dishwasher. However, if he does not generally do this, then he may not do so on Shabbat, as it would amount to preparing on Shabbat for the weekdays, which is forbidden.
One may not even turn on a dishwasher via a timer, because the dishwasher is designed so that it will not go on unless the door is closed. This means that the person who closes the door to the dishwasher after the dishes have been loaded causes it to turn on (SSK 12:37). However, in times of need, when there will be a great need to wash many dishes on Shabbat and it would be extremely difficult to wash them by hand, one may set the dishwasher with a timer. This is because causing it to turn on by closing the door is considered grama. Since an action performed via grama is not a true melakha, in times of need it is permitted (above 9:9; Responsa Me-rosh Tzurim §30).
If one can disable the system that makes the dishwasher’s activation dependent on closing the door, so that the dishwasher will go on at the set time even if the door is not shut, then even in ordinary circumstances one may load the dirty dishes and have the dishwasher wash them during Shabbat (R. Yeĥiel Faust, Le-ohavai Yesh 1).