Many communes have failed over this exact issue. They go on for several years with several people, usually women, making the sacrifice and doing what needs to be done. But eventually, the people who do the dishes get fed up and stop, or leave. And then the whole thing collapses.
The alternative is for the leaders to have some type of power to compel people to do the dishes, and some type of punishment to mete out.
There are certain tasks in society that no one wants to do. In order to get them done, you have to choose the carrot or the stick. Money or punishment.
If the society has no money, there's going to be a whole lot of punishment going on. It can work in theory, but in practice, a society based on punishment tends to snowball out of control, with the people in charge of punishment going too far. The people in charge make a small mistake in the size of punishment relative to the transgression. With money, small mistakes like this happen constantly, and they are constantly being adjusted by changing prices and salaries. The garbage man makes a higher salary than other manual laborers. But without money, the process has more steps, and is harder to get right. People protest, the leaders listen to the protests, go through the rule changing process, and eventually adjust the punishment to fit the transgression. Without money, more people are involved in the process. Some are removed from the actual issue. It takes longer for the adjustment to be made. With only the tool of punishment available, it's more difficult to fine tune every mismatch. In practice, it's really, really hard to get a punishment based society just right.