I disagree with many of the people here. You can do this and in a positive and reinforcing manor. It will take effort, and would be better from your Mom than from you, but it should still work just fine. It might not be worth the time if he is moving out soon, as its a slower processes, but one with lasting effects. Put the time in now and it will have a lasting outcome. One very hard component of this is that you must be sincere and never condescending. (it can be tricky to not be condescending as the sister but its important). You need to plan a "Prompt", "desired behavior", and a "Consequence"
You have to identify the behavior you want changed. Specifically. In this case, clean dishes after himself. The prompt is simple, there are dirty dishes around. When he cleans them up and washes them, you have to celebrate it this is the "consequence", not jumping up and down or anything silly, but make an honest and thankful verbal comment that he can hear and understand where you recognizing the behavior you want. (simple stuff, "Thanks for washing those dishes", "thanks for clearing those cups", etc nothing crazy but must be honest and sincere) Touch is often recommended with this, a high five, actual pat on the back, shoulder squeeze, something light and casual, tied to the positive thankful comment. (A tossed candy works as an alternative to the touch depending on your relationship, a Hershey's kiss, dove chocolate, or something similarly small you know he likes. You can get creative, there are plenty of none contact options out there.) It will honestly feel silly doing it at first, but over time if your consistent it will have the positive affect you want.
Ok, there is a problem, he never actually does the behavior you want in the first place. You need to use a tool called Simulation. This can be a bit harder as a sibling, being someones equal, or younger sibling its going to be a trick. However you can probably get your mom to help. Simulation is making a situation to simulate the prompt and behavior you want, so you can do the positive reinforcement. You can team up with your mom to have her make him do something, and you can add the positive reinforcement. Once the ball starts rolling it will cascade and grow.
I'm a big fan of "Everyday Parenting", the tools it teaches work on kids of all ages, and adults. I found the same practices that work on my kids work on my employees. It even works when someone knows about it, and knows your doing it. I had an employee train on it and now does it with her direct reports with positive results, and knows I'm doing it with her, but it doesn't mater. Because at its hart you are being honest and sincere, if your condescending, cynical, nagging, etc it wont work. Also, a side effect is that it has a net increase in general happiness and positivity.
This actually draws from some known aspects of influence/persuasion, the positive reinforcement falls under "Reciprocity", this only works if your consistent, its expected that your also doing the correct positive behavior which falls under "social proof". These tools work from all angles and can get far more complicated, but are worth the time and effort to learn and apply.