I'm type 2, but I have a similar issue. I was only diagnosed just under 3 months ago, and the foot burning started about 12 months before that. I've had my foot check and like you have full sensation, good circulation and classed as low risk.
However, despite what my DSN says, I'm certain that this burning sensation is some kind of nerve inflammation if not the beginnings of neuropathy. There are a few things that I have done that have helped, to the point where instead of the whole foot burning, it's now just mostly my second toe.
Firstly, I only wear shoes that are comfortable. Heels are out for now, and boots are in. I wear fluffy, light bedsocks all of the time (I bought several packs for a few quid off ebay) - even over my tights and under my boots! I've found the extra warmth prevents excessive burning and tingling... especially in bed.
The other thing I do is take an Alpha Lipoic Acid supplement. It's supposed to help repair damaged nerves. After a couple of weeks of taking it I noticed a huge difference.