The thread title gave me "fond recollections" of the effects of a couple of medications that doctors have prescribed for me "because they usually help something and everybody's taking them." The actual description doesn't sound much like that though, since with the common medications the "burning skin" is usually everywhere for susceptible people.
A few medications, particularly in the blood pressure and cholesterol regulating groups, can cause similar sounding side effects, and onset often doesn't begin to be noticed immediately; but the effect is cumulative with continued use. Niacinamide (Nicotinic Acid) is one that's fairly commonly prescribed, that builds up slowly, and is well known for the "burning skin" effect in susceptible people.