Death

Death happens when heroes fail at Death's Door: heroes struck while at Death's Door have a 33% chance to die. Heroes that suffer a Heart attack while at Death's Door will die instantly.

In The Crimson Court (DLC), a hero affected by the Crimson Curse will instantly die of thirst if he doesn't have The Blood at the end of the Wasting State.

When a hero falls, he cannot be recovered. It was the case before, but there is now a very low chance to have the Town Event From Beyond, which allow the player to resurect one of 3 random heroes from his graveyard.

Game Over
There is no real Game Over in Darkest Dungeon. Recruiting new Heroes is free. And because of that you can embark over and over again with new Heroes. By going in, collecting some Gold and then Dismissing the useless Heroes you can build your Gold back up, although Gold problems should be easy to avoid.

In New Game + however, there is an added failure condition where if the player does not complete the game in 86 weeks where the Heart of Darkness awakens prematurely or has 12 deaths as the deaths of the adventurers cause the Heart of Darkness to awaken from the blood of their deaths, they will get a Game Over and have their save subsequently deleted.