I had always thought that when somebody is all washed up, finished, past his sell-by date, a nobody etc, that the reason he is washed up is that he resembles the flotsam and jetsam washed up on a beach. He has been deposited by the high tide of fame upon the shores of obscurity.
I was wrong. The washing up here, is the same one that you do with dirty dishes.
You see, you wash up dishes once you're finished with them. Similarly, when an actor has finished his show he washes up i.e. he gets the make-up off his face and cleans the thespian grease from his hands.
The OED quotes the following extract from a 1925 article in the World newspaper, which, as the title doesn't suggest, reported New York stage gossip. Here an Italian performer with a trained canary is being fired.
‘That guy might be all right if he washed up,’ commented Buck.... Just then the stage manager called out: ‘What will I do with this act, Mr. Ziegfeld?’ ‘Wash up him and the bird,’ said Flo [Ziegfeld] and that was the last of the Italian and his trained canary... Hype Igoe, the World's sporting writer, heard of the incident... and in commenting... upon Frank Moran, heavy weight pugilist, advised that matchmakers ‘wash him up’. The phrase caught the sporting fancy‥and has become a colloquial fixture... as a meaty synonym for finals and farewell.
I can't find the full original, but I have conceived a sudden desire to start training a canary.
You're all fired.