Just a quick question to get this clear: can I use Dwarven Tomb to get a dead spirit hero back from my discard pile? Is a hero belonging to the spirit sphere considered to be a spirit card?

Heroes cannot be moved out of the discard pile unless the card specifically mentions it like Fortune or Fate.

"Note that hero
cards that are defeated are placed in their owner’s
discard pile. When resolving effects that move cards
from a player’s discard pile to his hand or deck, hero
cards in the discard pile are ignored, as hero cards
cannot move to a player’s hand or deck." (p. 20)

No. Unfortunately not. Even if you did get it back you cant play it because it doesnt have a play cost. You can only get it into play using a card that specifically allows you to return a hero to play.

Beast Rabban said:

Good point, we figured this was the reason, but overlooked the remark in the rulebook.