Normal painkillers won’t touch it - as you’ve discovered. A quick fix is to confuse your nerve endings by trying cool packs which will relieve it for a while but if the burning comes back then you may need to apply a hot pack which sounds odd but it may remind your nerves what hot really feels like and reset them. It sounds strange but I heard it on these boards and tried it in desperation once and it worked.
However, I had a sensory relapse 4 years ago where my left arm was burning so badly that nothing relieved it. I spent a week trying to sleep with one arm in mid-air (impossible by the way). It felt like I had ants under the skin and I was wearing vest tops in October as I couldn’t bear anything anywhere near it. My MS nurse (the local one) eventually got to me and prescribed amitriptyline which took another week to kick in. By that time I was zombie-like.
I’m still on amitriptyline now because if I try to cut down the itching and burning starts to return mildly and I just start taking them again because I’m not every experiencing that pain again.
If you’ve had this for a few days now, I suggest you ask for a neuropathic painkiller. Many of the ones used for MS (eg amitriptyline) are antidepressants but when prescribed at low doses they are used for neuropathic pain. A useful side effect is that you get a good night’s sleep too