Remove the grout from between the tiles. Use a utility knife to dig the grout out from between the tiles. Make several cuts per row, pushing the tip down into the grout to cut it free. When the blade stops moving easily, replace it with a fresh blade.
Insert the edge of the putty knife beneath the first row of tiles. Gently tap on the handle of the knife with the hammer until the knife is about midway beneath the tile.
Press upward with a firm, but light motion on the handle of the knife to pop the tile up from the vanity base. Continue until all the tiles have been popped up.
Place a chisel in the center of a tile and hit it firmly with the hammer. This will break the integrity of the tile and help loosen it from its base. Pick out out loose pieces.
Position the chisel at a 45-degree angle to the rest of the broken tile and hit it firmly with the hammer to remove the rest of the tile and clear a small section of the vanity top.
Insert the edge of a pry bar beneath the tiles that surround the area you just cleared. Exert downward pressure on the pry bar to crack and break the tiles free from the vanity top. Continue until all tiles have been removed.