If you run out of time on Crumbling vault, do you die?

I was actually wondering this myself too just recently when I encountered this event. It turns out that no, your character doesn't automatically die if you run out of time; you just can't get the bonus loot at the end of the dungeon, and you don't get the achievement associated with the event.

Here's a video I found that illustrates that nothing special happens when the timer runs out:

A quick google search leads to this wiki page which states:

If the player doesn't finish the event in time, a message will pop up informing that the hero has failed the quest. The player doesn't die, as was the original version of the event as seen in BlizzCon 2009.