Is it possible for a co-op game to work well without a timer or in some cases, a "death clock"?
Is it possible...?
Gears of War the board game doesn't have a timer system. You just die.... :\
I suppose as long as there's some way of failing other than just running out of time it should be fine.
It depends of your definition of timer. For example, shadow over camelot could be considered as a disguided timer. Both the swords for completing or failing quests, and the catapults accumulating at the door of the castles are somewhat timers without actually counting turns. But they actually increase according to certain actions that occurs every turn.
Not sure if what I said makes sense.
So, good games tend to have a sense of tension and progress. If you label all sources of tension and measures of progress as timers, then the answer seems to be no.
However, if the AI represents an alternate character in the game with victory conditions, like "have X amount of cash" and the AI gains cash through the actions of the players and random events, I wouldn't call that a timer. Cash here being a proxy for anything from health to strength to number of cattle. In that same setup, the AI could achieve a point of no return, but have no specific rule about it. Then you create the unpleasant scenario of the players cooperatively needing to decide to give up.
Death can be proxied as locked up, broke, or a number of other player elimination concepts.
Failing by death would probably be the only mechanic I can think of. When i say clock, i do also refer to alternate timers. For games like Forbidden island, the clock is the waters rising. With pandemic, there's multiple clocks.