I believe this is the quote you refer to?
Originally posted by Gatherer:
Rakdos's last ability won't reduce the cost to cast Rakdos itself. It applies only to creature spells you cast once Rakdos is on the battlefield.
And I understand the confusion!
Rakdos' cost reduction can only apply once he is on the battlefield, and that's what that ruling is trying to clarify.
However, when you cast a creature spell with Rakdos on the battlefield, his cost reduction looks at how much life was lost by all of your opponents during this turn. If you have 4 opponents, and cast Atarka's Command to deal 3 damage to each of them, then cast Rakdos, you could then cast Emrakul for 3 instead of 15. (4 x 3 = 12.)
For your original question - Burnished Hart will cost 3-3, or zero.
d:^D