Remove the top of the table from the legs and apron with a screwdriver or wrench as needed. Place the table top onto a workbench and secure it in place, with a clamp on both ends. The clamps will prevent it from moving as you remove the veneer.
Strip the first layer of veneer with a wide butt chisel. Hold the chisel with the bevel side up, and place the chisel onto the table top at about a 30-degree angle. Scrape off the veneer with the chisel, going against the grain of the wood. Work in rows, beginning at one end of the table and working toward the other end. Wipe the surface with a damp cloth to remove any sanding particles.
Remove the second layer of veneer using the same process as in Step 2, except work with the wood grain for the second layer, perpendicular to the first layer's direction. Wipe the surface with the damp cloth to remove sanding residue.
Sand off any remnants of the veneer or adhesive with coarse sandpaper attached to a sanding block. Remove the sanding residue with a damp cloth to complete the veneer removal.