I save some boxes if I think I might be able to use them for something else. Plastic containers, never would I get rid of them. I have all my shoes in either their original box or in a shoe size plastic containers to keep them from getting dusty. Same with purses I don't use often, in a plastic container so I can see what's in it.
I use shoe size plastic containers for fat quarters by color, a bit larger plastic containers for scraps by size. Plastic containers about the size of 12 x 12 or 14 x 14 for projects I'm working at from time to time. I use plastic bags for when I'm doing steps to a pattern, keeping all the small fabric pieces with each step, then have them hanging on a hanger to keep them off my cutting table.
I also use tin candy/cookie containers for my bias tapes and zippers. It gets so dusty down here in the basement so I need to keep items I don't use often out of the dust.
When I do get too many boxes and/or plastic bags I'll toss them, but never plastic containers as they cost bucks.