I like the concept, but I cannot think of a practical way to do it.
The only thing I can think of is to have an old
wood fired kitchen range with a
water heater, and plumb the sink to that, but that is really no different then any other means of heating the water and sending it to the sink.
I have a hard
enough time lighting the stove in the morning when it goes out; by that I mean I do almost anything to keep the stove burning, even if it is 50 degrees out, so I cannot imagine myself firing up a woodstove, just to warm water for a sink washing.
A kerosene smudge pot, or a coleman type stove might be more practical, but not all that "green"???