I usually keep the boxes, but it depends a bit. All boxes of BJD's I keep, because those are really handy for storing dolly clothes, props and stuff. Some boxes are just too pretty for even thinking about throwing away, like Fairyland Puki puki boxes. Also if I'd sell one of the dolls later it would be good to have the box around. I kept the box of my Dal Coco because it was pretty, but it isn't too good for storing anything... It is just pretty. I'm kind of sad that I didn't get the original box of my Pullip Lan Ake. I bought the doll second hand and the previous owner maybe didn't have the box anymore. :/ I would have wanted the box to go with my Dal Coco's box, but no can do. :/ Pullip boxes are pretty, so I'm pretty sure that I will keep all of those that I get.
The boxes of Monster High dolls I throw away. They aren't too pretty or handy, and the pictures in the boxes don't mean so much for me because I won't keep the dolls as they are anyway.