Is a mage with the vampiric trait unable to heal if he also has the finite trait? Or is vampiric considered different than heal (like reassemble is for the skeletons)?
Vampiric tells you you Heal which is affected by Finite Life.
Non-Living things cannot ever heal, but there is a similar ability called Reconstruct which a very few cards (5 cards in the game maybe? One of which is a promo).
All non-living targets already have Finite Life as standard, and can't be affected by Heal hence the Reconstruct keyword. Reconstruct can't affect the living. Ergo any given target can only be affected by one or other...