Yes, the Firebiner and Fire Escape are TSA-friendly and have flown all over the US and the world. The only problem we know about is China – can’t take any "sparking device" on a plane in China.
Unfortunately there is no official TSA approval process, but there are no requirements that would warrant these being confiscated. That said, if a TSA agent does keep your Firebiner or Fire Escape, take a picture, let us know, and we will send you a replacement.
If carrying on your person remember to take it off, or out of your pocket, and put it in a bin to go through the item screener.