Thanks for the responses guys! I've been thinking and I do like the idea, but I think it would have to implemented correctly. My main issue is thinking about how collecting time could become a mundane task. Has anyone ever played Dark Cloud? In Dark Cloud, when you go into a dungeon, you get thirsty so you have bring water with you. Whenever I see someone play this game, they always seem to get thirsty extremely fast, and they start losing what little health they have. Thus, they have to return to the surface and get more water from a shop. It gets very repetitive and boring. So, in a game where the player is always limited by time and they need to collect time to continue, it is imperative that the collection of time is an interesting task, but it can't be required so often that the player is limited by it. Just a thought.
Obviously, the player would collect time in various ways, and they could collect it from people. But could they collect it from defeated enemies? My gut instinct says no because it's time and not a tangible object, but then again, most games reward players for completing tasks and defeating enemies. Even on a more extrapolated level, I don't think the player would recieve time for completing missions unless they directly have to do with the acquisition of time! This would make the player work to create ways to obtain time, and if it is a sandbox style game, there would be hundreds of ways to do this.
You said that time would drive the player to create ways to obtain time, which could in turn be rewarded for archetypical behavior. I think this idea is good, but I also think that the idea may have a serious limitation. In order to create and maintain the adrenaline rush and the fear factor of death, the amount of time the player is rewarded for completing tasks would have to be very small, I think somewhere on the scale of 20 minutes or some value that would keep the player's time relatively low. Of course, the player will have spurts where he accumulates large amounts of time, maybe even years at once, but gameplay elements would force him to use it up. I can imagine what kind of fear a player would get when he doesn't quite calculate right, and buys an item at a shop and is left with only five minutes!
I totally get that, I'm mainly just throwing ideas out there right now to see others have the similar thoughts. I won't even think very hard about designing a game based on the idea of time as currency until summer (if ever).