What are some ways to get rid of mattresses/box springs?
I have two Sealy Posturepedic queen size mattresses, box springs, and frames. The beds are 10 years old but in excellent condition. If I can’t sell them, which charitable organizations pick up beds? Or, do the trash collectors pick them up? I live in an apartment complex and was told I can’t put them near the dumpsters.

Try freecycle.org. You offer them for free and whoever you agree to give them to will come to pick them up. Works great.
Yes good will takes mattresses and box springs.You can call around to charitable orgs.
Cut the covers off and use large bolt cutters to cut the wire springs up. Then toss the metal in the recycle bin. Just cut a mattress up recently. Took about an hour to get it all done.
You have a telephone, so use it. It may be you may have to haul them off to the dump yourself(disease prevention) Or you could put up a notice in the laundry room billboard and see if anyone else is interested in them and maybe they can haul them out into their unit.
put them on craigslist for free if u cannot sale them, call a local church, thrift shop etc