Is there anything I can do about slightly melted carpet from a hot iron?

You could mow the entire apartment. Or maybe you could find a square in a closet and do the old square switcheroo patch job. You would probably need to do an extremely good job for it to not look worse, however.

you can have it matched and replaced, a little costly, but I can promise you that your going to be charged a lot more by the place your renting from if they have to do it….....

Tell the truth: “I’ve never ironed on a carpet before, but I’m
young and don’t know much about housekeeping and I’m
trying hard to keep this place clean and my clothing spiffy; but now this has happened and
(besides the fact that YOUR CARPET has plasticked up
my favorite shirt) I intend to correct my error. Please find the exact carpet description for me. I will have an installer come out at my expense and patch this.”
The landlord will never want to let you go, will write you excellent references when you move on, and will bring you over little packages of brownies from time to time. S/he may even buy you and ironing board. I’m a landlady and I would.

How on earth did a carpet melt?

I have lived in lots of apartments and, while that [letter above] may work with some landlords that only rent a couple apartments out, I don’t think that it’s going to work on the leasing office for an apartment complex, if that’s the kind of apartment were talking about…...
The last apartment complex I lived in ($1700 a month) and they wanted to charge me $800 because there had been 2 pictures hung on the wall and 1 stain on the carpet in my roomates room…

I didn’t mean that in a rude way, I apologize.

buy a new one! or put some furniture on the destroyed place.

if she moves how will she hide it?

hmmm, yeah thats a tough one, the only way I can think of you getting out of this is if they are already planning on replacing the carpet after you leave, they usually do that every few years. But then you have to hope they wont be shady and charge you for the damage anyways.
The best way though is to get a replacement patch from who ever made the carpet, or the closet I suppose, (but then what about the closet?) and just patch it up, if you are very detailed with it, it should look okay. the way you make the seam disappear is to pull out the carpet strands from some where else, like another replacement patch or something, and just glew them where the seem is visible, if you do this well enough you shouldnt even be able to notice the difference.
http://www.ehow.com/how_113128_remove-burn-marks.html

Is the same carpet in a closet? You could cut out a patch from a far corner in there and replace the part you melted. Just make sure you measure both both pieces do they’re the same size.

@melon…most cheap carpet has “plasticky” stuff in it. Some clothing fabrics melt as well.

People!People!
whatever you do with destroyed carpet, it is still destroyed, so tell that you have destroyed caroet, buy new one and take a bad one home! :)
there is not good way to “hide” melted part….WHATEVER YOU DO…

Though cheeb recently called me a whore in public, I have
to bow to her giant brain on this one.
Repair advice addition: hire a professional to find a hidden
piece to patch the melted place with, invisibly patch
THAT spot with something semi-matching, and put the perfectly-matching new patch into the burned spot really beautifully.
Cheaper than what a venal management company would
do to you.

when did I call you a whore?

O gosh, long long ago. It was a question about whether it was nice to sleep with lots of people. I said yes. I’m not mad at you.
But I’ll continue to bring it up because I found you quaint.

Try to lift the pile with the use of a wire brush. But don’t go mad or it’ll look too fluffy.