Question:
Where in the Bible does it suggest that men should wash dishes?

Answer:
I am not aware of the Bible telling us about men washing dishes, but here are verses that tell us how men wipe dishes—which may mean that they dry them after they are washed:

“Therefore thus saith the LORD God of Israel, Behold, I am bringing such evil upon Jerusalem and Judah, that whosoever heareth of it, both his ears shall tingle. And I will stretch over Jerusalem the line of Samaria, and the plummet of the house of Ahab: and I will wipe Jerusalem as a man wipeth a dish, wiping it, and turning it upside down” (2 Kings 21:12, 13).
In the verses above the LORD says that He will judge Jerusalem and Judah like “as a man wipeth a dish, wiping it, and turning it upside down.” It is interesting that when a man wipes a dish he turns the dish over so he can look at the side of the dish he is wiping. It seems that women don’t do this!
This way of wiping a dish shows that the Lord knows precisely what He is doing when He judges, and especially so when He judges His beloved earthly people Israel, for He addresses Himself as “the LORD God of Israel.” We see this carefulness in the detail He gives throughout Revelation chapters 6-19 where He tells precisely how He judges this world in the soon coming day.