You mean like literal death or metaphorical death (i.e: game over but the character is not necessarily killed)?
Most horror games tend to have some gruesome and detailed death animations as a reward for your incompetence. Dead Space, The Evil Within or modern Resident Evil games are an example of it.
Prince of Persia (Sands of Time at least) had that unique piece of dialogue from the Prince whenever you get killed. He was like "ummm, yeah sorry, that's not how it went" or similar pieces because the game itself is narrated through the (living) Prince, so dying was quite contradictory. On the same vein, Fahrenheit/Indigo Prophecy ended the game's narration whenever you triggered a game over, and his narration had actually something to do with how you screw up in those final moments (Like "I was too slow to escape police and I spent the rest of my life in prison").
