I hesitated to use the word “fumes”, but I’m really quite certain that’s what they are. I just got a new 8’ x 10’ rug from rugsusa.com and it’s spreading intense fume smells across my whole main floor and into the second story. I’ve had a fan blowing over it for the two days that I’ve had it, and have opened all the windows when possible. I’m concerned about even letting my kids play around it because it’s just that kind of smell that feels like I’m inhaling a bucket of concentrated VOCs. I’ve had new carpet put in my whole house before, and I’ve gotten half a dozen new rugs in the past, and none of them have been anything like this. I’m considering renting a steam cleaner but that’s a lot of time and money that I would hate to put into a brand new rug.
I know this feeling well - you just feel uncomfortable even breathing the air in the vicinity. I had a friend who only ever bought 2nd-hand furniture for this very reason.
Do you have an outbuilding, garage or attic where you can leave it unrolled for a few days until the worst of it has dissipated?
You could even perhaps think about putting it in a car while it airs - at least you can open the windows and drive out the smell when you need to use the car!
The best way to get rid of smell of rugs, blankets, mattress etc. is to let them breath in the air and sun (of course, put it backwards so the sun won't destroy the main face of the rug).
If you can spray it with some freshening/good smell of some spray, that is good and won't hurt that kind of fabric/material - then do it as well.
You can also try rug cleaners materials before taking it out. If you don't have a place where you can leave it outside for a day or two try any other place with moving air, such as Lefty suggested here.
You can try leaving a few cups of normal vinegar in close vicinity of the rug for a few days. It might help quite a bit. Vinegar is also very suitable leaving in a room that have had smoking visitors (provided you don’t like the smell of smoke)...