I have had it going down my arms when I had cubital tunnel syndrome. Burning sensations often have to do with neuropathy, which can have numerous causes. If this is a new symptom for you and lasts beyond a few days then I would be sure to mention this to your doctor or schedule an appointment. There are definitely several helpful medicines too which can nearly eliminate the pain (or in my case turned it into a cooling sensation running down my arms instead of pain or burning which I much preferred!)
I have had paresthesias (burning, pins and needles) in my lower legs and feet, mainly. I also have a couple of patches on my back which burn and itch sometimes (like now). The burning has calmed down since I have been on steroids and I mainly get it now late in the evening or when I am particularly fatigued.
When they were particularly bad in 2006 I could not calm them down and resorted to ice packs. High steroids were the only treatment which could stop it for me.
I don't know whether this is typical for lupus or a neurological overlap (mine seems to be neurological).