Getting out of a horrible place with minimum damage is a perfectly fine objective in any horror story with characters who don't have the obsessive motivation and/or power needed to fight back. Since achieving this objective has to be a long and nontrivial process (or there would be no story and no horror), key & lock puzzles are a natural design pattern to make player characters explore the place they want to run away from.

Key and lock puzzles as an excuse for horror moments become boring only if there is no development; a common but good resolution is accumulating resources, information and motivations as a byproduct of exploration and encounters until they cause an interesting change of purpose or strategy (a climactic fight, surrendering to death or self-sacrifice, realizing who the real enemies are...) to end the game or story.

I suppose that, as ActiveUnique perhaps implies, the problem appears when key-hunting is the most prominent element of gameplay in a large number of games. So perhaps an idea might be to consider alternative gameplay mechanics.

First of all, the ubiquitous fallback position of action elements--be they shooting, hack-and-slash or whatever--are in most cases probably not a good idea for most horror games: action elements empower the player, which can decrease fear, thus hamstringing a horror game.

That said, action might work if it's risky, or if the game is a roguelike/lite, however.

Puzzles and other logic-based mechanics might work, but I do worry that engaging the player on a logical level might reduce their emotional response--but I'm not sure of that.

One gameplay type that seems as though it might work rather well is stealth: the idea of hiding from and avoiding horrible things seems quite natural. Just look at some of the horror-themed levels of the Thief games.

That's what comes to mind offhand; what other ideas are there?