Hi Cynthia! Such a great question, I'm sure a lot of girls have this problem. Although I store all my shoes in my shoe closet, I am the same as you and keep all the boxes. My solution was to put up shelves in the corner of our closet (we are lucky enough to have a large walk in closet) and stack the shoe boxes on the shelves. If you don't have extra space in your closet, how about just putting shelves up in a corner of your bedroom and stacking them on there? Living in a small space, J and I try to utilize as much wall space as possible and are big fans of these floating wall shelves at Ikea.
Another tip for the shoe boxes is that I felt it was a bit silly to just have these empty shoe boxes taking space so I decided to use the boxes as storage. They hold things like buttons and thread, shoe bags, others hold my crafty stuff like ribbons and pens and one box even holds my bikinis! It's a nice space to keep all those things where I just don't know where to keep them and since each box has it's own "category" of items, one peek in the box and I know if it has what I need.
If anyone else has tips for Cynthia, leave them in the comments. Would love to hear your ideas as well!