If you have an old mattress and box spring in good condition, consider selling them or giving them away on a web site such as Freecycle Network, Nextdoor or craigslist. If you purchased a new mattress, see if the store will take your old mattress for recycling.
You can also check with your local recycling center or garbage disposal service to see what their rules are with old mattresses and box springs. Garbage haulers will often pick up a certain number of oversized items each year. Yet another option is a junk removal company, like 1-800-Got-Junk?