Lay them out in an open place to dry....then dispose of them like any trash.
You need to be careful not to close solvent soaked rags in a container because there is a danger of spontaneous combustion.
Linda C

Ditto what Linda said. Long ago my ex left a pile of tung oil soaked rags on the floor in our den. We woke in the night because the smoke made us cough. We found a smoldering mound of rags. Very scary. Now I spread them out on cement to dry.

At the end of the day, I usually wash them out in warm soapy water, and wring them out. Then I open them up and lay them across my saw horse to dry. If there is still a lot of residue, I put them in water till they are soaking wet, put them in a plastic bag, and throw them in the trash. The used paint and lacquer thinner I usually strain through a coffee filter and keep it in a coffee can to use again, like the first stage of cleaning a brush or cleaning off paint remover.
You asked how we did it, not if our way was the best.

Thanks for the suggestions--I appreciate it.

SInce you were all so nice about my initial question, I have a few follow-ups:

Gloves: do you use decent ones, rinse and dry them, or use cheap disposables and discard every time? And can they be dried and discarded in the garbage?

If you take a solvent out of its container to put in a smaller one, what do you use? Glass and re-use it? How do you dispose of it?

I've been using a mix of turp and linseed oil to freshen up some furniture. I put it in a cottage cheese container and now don't know if it's safe to keep it in there.

Sorry if these questions seem dumb/basic. I'm super-paranoid about fires because we had a housefire once, and I want to do the right thing in terms of environmental safety.

I use "refinishing gloves"...clumsy but won't dissolve in paint remover, and since they are expensive Is ave them.... I return clean solvent to the large container....and a linseed and turp mix would be fine to save....but it won't keep very long...but won't spontaneously combust.
Linda c

Thanks, Linda!