Strip the floor down to the underlayment using the technique required for your current flooring.
Place a piece of underlayment and a piece of your vinyl tile on top of each other. Set the underlayment and the tile next to the door jamb. Mark the height of the underlayment and tile onto the door jamb. This is where you will make your cut.
Saw through the door jamb and casing using an undercut saw at the mark made on top of the tile and underlayment. Remove the wood from the door jamb.
Attach the underlayment to the floor underneath the door jamb using the correct nails for the flooring you are attaching the underlayment to. Remove the paper from the adhesive backing on the tile, and attach the tile to the underlayment under the door.