Locate all of the upholstery tacks on the wood furniture. These are likely located anywhere that the fabric is attached to the wood, such as along the sides and on the back.
Slip the edge of your needle-nose pliers underneath the first tack on one end of the wood. This should loosen the tack, but don't force it or you risk damaging the wood.
Use something slimmer, such as the blade of a utility knife, if you cannot get the needle-nose pliers under the tack.
Pop the tack off the wood using the needle-nose pliers.
Remove all remaining tacks using whichever of the two methods that worked best for you.