What would happen if you tried to wash a linseed-saturated c
It takes HEAT to start a fire.
If the linseed oil soaked rag was fresh, and the oil had not started to heat up and form a hard layer, then I suppose enough of a good detergent and hot water could not only solubilize the oil and allow it to be rinsed away, but also reduce the local temperature of the oil/rag and prevent ignition. The water would also prevent oxidation to some extent. It would probably be messy, though, like trying to wash an oil-based paint soaked rag in water. Not the best combination.
But I think once you've dipped a rag in linseed oil it's probably done. You could try to wash it out in mineral oil, or paint thinner, like a paint brush, but since it's just a rag I'd let it go (in a metal container).