Remove the door from its hinges using a screwdriver, then set the door over a pair of sawhorses so that the splintered side faces up.
Squirt wood glue directly over the splinter so the glue gets in around the splinter's length.
Place a piece of waxed paper over the glued splinter. Lay a small, flat piece of wood over the splinter. Place a 20-pound weight on top of it. Wait until the glue dries.
Remove the weight, wood and waxed paper. Sand the splintered area smooth with 150-grit sandpaper. Refinish the door.
Lift the splinter up with a pair of pliers until it snaps off. Cut away any broken wood fibers protruding above the hole in the door with a utility knife.
Using a putty knife, scrape out wood filler the size of the hole in the door. Put it in a disposable plastic bowl. Squirt a 3/4-inch bead of wood-filler hardener onto the filler, then mix the two together until blended.
Scrape up the filler and press it into the hole with the broadside of the putty knife. Apply enough filler to fill the hole about 1/8-inch above the door's surface. Wait 24 hours for the filler to cure.
Sand the wood filler smooth with the surrounding surface of the door with a 150-grit sanding block. Refinish the door.