What’s the best way to remove wall anchors or nails without damaging the wall?

Place a stiff putty knife or a thin piece of wood/cardboard flat against the wall next to the nail head. Position the claw of a hammer over the nail head, resting the hammer's head on the putty knife/wood. Gently lever the nail out using the putty knife as a fulcrum and protective barrier. This prevents the hammer from denting the wall.

For screw-in plastic anchors, simply use a screwdriver to back the anchor out counter-clockwise. For expansion anchors (that spread behind the wall), first remove the screw. Then, try gripping the anchor's lip with needle-nose pliers and gently wiggling and pulling it straight out. If it resists, you may need to push it slightly into the wall or carefully cut the lip flush with the wall and spackle over it.

If a plastic anchor won't come out with pliers, select a drill bit slightly larger than the screw hole but smaller than the anchor's outer diameter. Carefully drill into the center of the anchor to break it apart or weaken its grip. You might then be able to pull out the pieces with pliers or push the remainder into the wall cavity. Patch the hole afterward.