There are two issues here:
1/ The outers, or case boxes that the figures come in. These are generally of poor quality and very little value as their primary job is to protect the figure inside. The only reason I would consider storing one of these is for the product/case/batch details etc on the box. The box itself is unlikely to have any value at all. And should you ever move and need packing boxes the removal company usually supplies these or pop down to your local supermarket and pick a few up!
2/ The second issue is the inners, that is the display box the figure is in or on. If you are a MOC or MIB collector then you will obviously keep these but if you like to display your figures then you can either trash them or keep them. Personally I would keep them.
Why?
Because at some point you may want to sell them, as already pointed out, or you may have to move. Another reason is that you may wish to "retire" the figure, or give it a rest from display and then you will need to store it safely...so it's original box would be ideal to use for this.
A few posters have mentioned selling them on eBay, well to be honest I can't see anyone paying $10 for a box, plus shipping etc, and don't forget you will have to pay the sellers fees even if it DOSEN'T sell, but if you are selling the figure then you will likely get a better price if you have the original packing to go with it.