Distortion and overdrive is done by clipping the signal. So there really isn't a way to recreate this exact sound without some kind of electronic device altering the sound. This article shows some simple wave forms and what distortion looks like.
Resonator guitars still sound very similar to regular acoustic guitars. The way they resonate is a little different which changes the sound slightly, but not nearly enough to be confused with an over driven guitar.
Banjos essentially have a snare drum as the body and having snare wires would add a bit of a 'raspy' and 'buzzy' sound, but still not the same as clipping the actual guitar signal.
About the closest thing I could think of to actually distort the guitar's signal naturally would be to set the action so low on the guitar that the strings 'buzz' against the frets...but again, this sounds nothing like a overdriven electric guitar and just sounds horrible! lol
What goal are you trying to achieve? Is there a reason why you either don't want to or can't use an amp? If this is for a campfire/outdoor setting where you don't have access to electricity, there are small battery powered amps. However, I've never used one or heard one so I can't speak to their quality (I would imagine they're not great).