Youíre probably going to be tempted to tap the old finishing nails out from the back side, but thatís not a good idea. The heads of the nails will splinter the face of the molding as they push through. Instead of doing that, take a pair of locking pliers and pull the old nails through from the back. You can use a block of wood under your pliers to give you extra leverage.
Once you have all the nails out, youíll need to fill the holes with wood putty or you can just reuse the holes when youíre nailing the molding back in place later.