I see some people like to leave a big cake of dead rosin on the fiddle. I've heard talk of "the old-timers' fiddles looked like that." Well, they looked that way because the old timers were too damned lazy to clean the stuff off!
It's made out of tree sap. Turpentine's made out of tree sap as well, I believe. The rosin can act like a solvent and will eat the varnish.
I'd think wiping the bow with a dry cloth ought to get the excess rosin off. You don't want to take it all off. Washing it might work, but in my opinion, it's overkill. Works better for cleaning the hair when someone gets hand-slime on it because they don't know not to touch the hair, or spills some of their greasy dinner (or Guinness!!) on it.
I've noticed that rosin sometimes starts clumping (minutely) and makes the bow sound grainy. I've scraped the stuff off with a blade. I don't know that this is such a good idea because it's possible to shave off the little barbs on the hair that cause the rosin to stick in the first place. I haven't had any problems doing it, though.