At first glance, there would not seem to be fundamental problems with operating the machine on a timer, as you are not changing anything of Shabbat significance by your actions on Shabbat. After all, the same activation of the electric device and the heating of the water will occur regardless, whether or not you fill the racks with dishes. The removal of the grime from the plates by means of the hot water (which occurs only because you put the plates in on Shabbat) is not considered borer[1] or bishul.[2] (The reasons for this are beyond our present scope.) It is likely that the detergent will become cooked, and it should therefore be put in before Shabbat (also, see below).[3]
However, the halachic problem arises as a result of a common safety device. In order that hot water should not escape from the dishwasher during its operation, the systems are designed so that the machine works only when the door is closed. Obviously, when one loads the dishwasher, the door is open, and then he must close it in order for the machine to be activated later by the timer.
The impact of the closing of the dishwasher’s door on Shabbat is classified only as gerama, an indirect causative action. It is less severe than a direct chain reaction, like pulling the trigger on a gun, or even a delayed chain reaction that takes significant time to develop, such as changing a timer to go on in an hour rather than in ten hours. Here, the closing of the door simply allows a separate process, triggered independently by a timer set from before Shabbat, to be impactful later. This gerama is far from a full violation of Shabbat and is permitted in certain situations that warrant a low-level violation of Shabbat.[4] However, in most cases, it is forbidden to cause such a delayed-reaction melacha. For example, we do not allow one to press the buttons of an air conditioner that will make it go on later when a timer activates the electricity.
There is a technical solution, albeit a problematic one, that one can arrange with the help of an appliance technician to bypass the aforementioned safety device. (The Zomet Institute provides this service.) Then closing the door has no effect on the dishwasher’s operation. (One caveat is pertinent. The detergent should be placed in the body of the machine, not in its usual place in the door. Otherwise, it gets into the washing cycle by virtue of closing the door.) Unless one can ensure that this will not cause dangerous situations (such as opening the door during operation), we would say that this is forbidden because “danger is more severe than prohibitions.” However, we cannot preclude the possibility that someone can create safeguards that make such an option a responsible one.
As far as using the dishwasher a second time, we know of no way to have a second cycle without an act of gerama, such as pushing a button, turning a dial, etc. If one preferred to have that cycle take place on Shabbat afternoon for use after Shabbat, rather than Friday night, this can probably be done with the safeguards in place. Generally, it is forbidden to prepare on Shabbat for something needed after Shabbat (hachana), even if the preparation does not include a prohibited action.[5] Although filling the racks with dishes is an action that takes place on Shabbat, it need not be considered preparation, as many people find the dishwasher to be a good place to temporarily store dirty dishes, and closing the door is natural for aesthetic reasons.
A final issue, which may or may not disallow having a dishwasher go on, is called avsha milta. The Rama[6] forbids having a mechanism operating from before Shabbat that one is forbidden to operate on Shabbat, if it makes noise (which might give others the wrong impression that he activated it on Shabbat). The issue is relevant even if the mechanism goes on by timer during Shabbat. It is permitted only if it is common for people to set up the mechanism in advance and there is therefore no reason to suspect that one desecrated Shabbat.[7] This could be a problem for a dishwasher. Rav Moshe Feinstein[8] says that the forbidden level of noise is such that it can be expected to be heard in the next room, which is border-line for a dishwasher. While Rav O. Yosef[9] and Rav N. Rabinowitz[10] say that avsha milta applies to washing machines, relatively quiet dishwashers would avoid this issue.
In short, although in principle there may be a way to use dishwashers on a timer on Shabbat, a combination of technical and halachic problems makes it not simple in practice.