Ok ..
Take the unglued peice of Lino and match it up to the pattern on the floor over the burn hole.
Tape it down with duct tape or masking tape.
Cut back to the nearest grout line ( if its a ceramic tile pattern)
Cut through both layers at the same time.
Following the tile all around the grout line.
Remove all glue left on the floor.. and replace the new piece with some glue on the back.
Roll with a rolling pin.
Use the seam sealer recommended for your floor.
And you are done. :)