Slip a putty knife between the backing and the nails and use it to pry.
If the backing fails then you can remove the nails using a pair of pliers but you'll need to find a new backing.
This of course assumes the person who assembled the furniture originally didn't use glue on all of the joints like I do. I've found that if you glue all of the joints in a flat pack when building it the expected lifetime can be improved by an order of magnitude or more, especially if you're moving it around regularly. Seriously it is night and day between the wobbly one you only followed the directions on and completely solid one that's fully glued.