Oily rags, used to apply oil or oil/varnish blend finishes, may be the most dangerous item you have in your shop because they can spontaneously combust and burn down your shop or house if left in a pile or thrown wet into a trashcan.
The easiest way to deal with these rags if you work alone is to hang them over the edge of a trashcan or tabletop, or spread them out on the floor. Don’t pile them one on top of another. They must be exposed to air on both sides so they can dry out safely.
If you are working with others, have everyone immerse their rags in a pail of water, then take them out at the end of the day and hang them individually to dry or hire a specialized company to dispose of them.
Once dry (and hardened) the rags are safe to throw in the trash. At this point they are no different, for the purposes of toxic waste, than wood with a dried oil finish.
Because manufacturers of oil stains are changing formulas to comply with VOC laws, rags used with these stains can sometimes also spontaneously combust. It’s wise to play it safe and treat these rags the same as you do the rags used with oil finishes.