I'm going to give you the same answer I gave to the question about going to Michael Bay movies: just stop doing it. Man, I'm lucky someone asked such a bizarrely appropriate and seemingly unlikely question on a site about dating advice, or this whole paragraph would have fallen apart.
But yeah, don't pick up after your guy if you don't feel like doing it. Because doing it when you don't feel like doing it, or feel unappreciated for it, isn't helping anyone. He'll learn that it's okay to let his ability to keep from wallowing in filth atrophy entirely, and you'll become a bitter, seething demon of resentment. Then one day they'll find him dead with a dirty sock stuffed into his mouth.
Tell him earnestly that you need him to help out more. From that moment on, just clean as much as you want to...your stuff only, if that's what it comes to. Yes, it will get filthy before it gets cleaner, but steel yourself. This is a dirty, horrible marathon, not a sprint.
Ideally, he'll come to his senses and get on the ball. If he doesn't, he's showing that he doesn't really have a lot invested in this relationship. It's not a cleanliness thing; it's a respect, consideration, and willingness to integrate the other person into your life thing. If washing a dish is a dealbreaker for him, you're better off not making the deal.