I've heard it but with the word "with": "Are you done with washing hands?" (It sounds odd probably) "Are you done with your homework?" (that's better).
"Are you done with your homework?" is fine.
"Are you done with washing your hands?" is OK but the "with" is unnecessary.
I would expect to hear either "with" or the verb+ing.
Are you done doing your homework?
Are you done with your homework?
In BrE, I would use "Have you finished your homework [yet]?" and "Have you finished washing your hands yet?" (or "Have you washed your hands yet?" which has a slightly different meaning.)
Bear in mind that "to be done with something/someone" can mean "I have had enough of it/you and will have nothing to do with it/you from now on".
My boyfriend had three affairs last year. I'm done with him. I'm going to find myself a decent bloke.
My boss just yelled at me again so I shouted "I'm done with you and I'm done with this stupid job". Then I cleared my desk and went to find a better job.