Sorry...I see a throw rug in your future.

doubtful...put a plant over it.

If you have extra of the flooring you can probably fix it- carefully cut out the affected "tiles" and glue in replacement ones.

Otherwise, well, if you are good at faux painting, you could try to paint and blend in the burns, making the deeper areas look like rock cracks... Probably put polyurethane over the paint once dry to protect it.

That burn doesn't look as bad as Paige Davis in pok-a-dot red heels. Please girl, call a wardrobe stylist. Sorry everyone, but I've been thinking that all day.

Ask This Old House did a repair similar to this recently (maybe last season). I can't find it on the website, but perhaps you're more patient than me with the searching.

Mazeppa - The dots at least draw your eye away from the new hairdo(nt).

If those are vinyl tiles, you might try prying up the damaged ones (use a hairdryer to soften the adhesive) and swap them out with less visible tiles, like the ones under the stove or refrigerator. If it's sheet vinyl, you could still swap out a piece cut from an inconspicuous spot with the burnt spot, you would just have to be more careful to cut the vinyl to an exact fit.

These may or may not help:
http://www.thisoldhouse.com/toh/article/0,,216707,00.html
http://www.naturalhandyman.com/iip/inffloor/inflino2.html
http://www.hometips.com/content/content_vinylfloor2.html

It looks like a heart, just paint it in red :)