Ideally, meeting the volunteers halfway would benefit everyone — but those who fail to respond to passive-aggressive postings reinforcing quaint gender roles may claim not to realise the message is intended for them.
Someone who leaves a dish to soak, then comes back to find it in the dishwasher, may even infer that using the sink as a way station meets the implied standard. But what if the snark were supplanted with something simple and explicit, such as "Please do not leave dishes or utensils in the sink at any time"? No one can pretend to misunderstand that.
If a crystal-clear request has been made and ignored, then it's time for Operation Why We Can't Have Nice Things. If your employer is supplying the kitchenware, the employer has the right to take it away. The employer also has a right to prohibit storage of unclean personal items in common areas, and to set a policy that any such items will be thrown out at regular intervals to avoid creating a friendly habitat for vermin.
And, of course, the ad hoc cleaning crew is always entitled to go on strike. In fact, depending on management's willingness to help arbitrate and enforce a clean-kitchen policy, that may be the best place to start.
The message is something anyone working for everyone at an environmental organisation should be able to get behind: If everyone doesn't make an effort to take care of our shared space, what's left won't be pleasant for anyone.