How much damage does burning do over it's lifetime? Does it differ depending on class or the pyro's weapon?
Assuming you mean Afterburn, which is the amount of fire damage you suffer after the flame thrower/flare's initial damage, assuming it is no longer directly hitting you:
Degreaser is 2 damage/half second for 10 seconds (40 total if not extinguished). If a player is jarated, it does 3 damage/half second.
All other flamethrowers, flare guns, lit arrows, or the Cow Mangler 5000's charged shot are 3 damage/half second for 10 seconds (60 total if not extinguished). If a player is jarated, it does 4 damage/half second.