Maybe an item that would look good sitting there, like a hamper or something I guess. Unfortunately unless you have a piece of the same vinyl hanging around, a patch cannot be done.

check with your homeowners insurance it should be covered...good luck

i've never tried this, nor do i know that it will work, but since it seems there is no answer but to tear the entire floor up or cover it with another vinyl. you may be able to cut out a piece of vinyl that matches that pattern , from an inconspicuous area, like under the fridge. you will only be able to get up the top coating like scoring and peeling a sticker off of cardboard box. cut out the damaged area to match the piece cut by laying the new piece over it. then just glue it down with vinyl adhesive and add seam sealer to the edges. seams will be noticable but better than a large burn mark, give you a little longer life out of the floor untill you can afford to replace or care to.