At home - use the powers of your microwave (assuming your scrub doesn't have metal content) to heat/steam sanitize an older scrub.  Let air dry and then put in small ziplock.

At camp - use small amount of bleach in your rinse water.  When done with dishes, scrub goes into the bleach water, and is wrung out.  Ours typically goes right back in the ziplock.  I guess you could air dry it out, or even put it on a hot rock next to the fire for a full dry.

Throw out with the trash upon return.