Remove any molding from the bottom of the walls, where the walls meet the floor, with a pry bar. Locate the nails that secure the molding to the wall. Insert the tip of a small pry bar into the seam between the mounding and the wall, just above a molding nail. Position a small block of wood between the wall and the pry bar, then carefully pry the molding away from the wall. Pry the molding away from the wall in small increments and work along the entire length of the molding strip to avoid breaking the molding. Do not forget to use the wood block to avoid damaging the wall.
Cut any large tiles into strips approximately 6-inches wide with a utility knife.
Insert the pry bar between one of the cut seams, where the a tile meets the wall, then lift the edge of the tile. Grasp the tile as close to the floor as reasonably possible to minimize tearing.
Slice torn pieces into smaller strips, then peel away as much of the vinyl as reasonably possible. Chances are good that the felt backing of some tiles will not lift off the floor. Loosen the backing by spraying a mixture of water and liquid dishwashing detergent onto the backing with a spray bottle. Allow the mixture a few minutes to soak into the backing, then peel the backing off of the floor.
Scrape away any small vinyl pieces and remaining adhesive from the floor with a floor scraper or wide bladed putty knife.