Burning sensation on upper back

I’m sorry to hear about this and this is certainly not normal. One important question first, however, did you see a fully licensed acupuncturist? or some other professional (PT, chiro, physio, etc.) who does “acupuncture”. If you saw another type of professional, you should see a fully trained and licensed acupuncturist and this type of poor treatment will be unlikely. If who you saw was a full Chinese Medicine practitioner, you should see someone else. While an occasional bruise from a needle happens and is somewhat common and generally harmless, bruising from multiple needles during a treatment, particularly a first treatment, is just bad technique.

Even from bad technique, however, problems from acupuncture are rare. So the burning you are feeling is probably just from the bruising and it should pass fairly soon. There is a possibility for nerve irritation from far too aggressive/deep needling (not common with fully trained acupuncturists in countries with licensing) - but even that will heal in most cases it just takes longer; possibly up to 12 weeks.