There is a lot more to be done with action in a game than just combat. The most common go to is vehicle chases - or even on foot chases - but even that is kind of limited. Think of movies, books, shows, and every other form of media. There's lots of things out there that can be used to add tension. The problem is, most games aren't set up to really handle those things. Or are they?
7th Sea vs. The World
Perhaps not surprising this is another post inspired by how 7th Sea does things. 7th Sea doesn't have rules for combat, it has rules for Action Sequences, and Combat can be done as an action sequence. This works out well because unless a villain is present you don't even roll dice for the opposition. However, if 7th Sea can use their generic action sequence rules for combat, we can use combat for action sequences from other games, right?
Hit Points Are A Hell Of A Thing
In D&D hitpoints aren't some magical number showing how many times you can be hit, despite what it sounds like. When you lose 10 hitpoints to an attack with a sword you didn't necessarily get stabbed with the sword. Instead, hitpoints represent your ability to turn lethal blows into less lethal blows. So maybe you didn't get stabbed, but the block rattled your shoulder, or you got nicked on the leg, or something else happens.
To put it more bluntly, your hit points are your stamina. When you're low on hit points you've barely got anything less in the tank and so a lethal blow can't be turned aside and can in fact kill you.
Now, I was talking about D&D but this can be true for other games. Even games like L5R and World of Darkness where you have wound penalties are using an abstract system. It's not like an Earth 4 person needs to be stabbed in the heart more times than an Earth 1 person. So even with the penalties, your number of wounds can be an abstract for stamina and mental clarity to keep fighting.
Why does this matter? Because hit points bring in a natural lose condition of exhaustion, passing out, or death as needed, and they're what systems are designed to work with.
Climbing A Mountain Can Cost Hitpoints
Consider the following situation. Sarah's thief needs to scale a mountain quietly while guards from the Evil King patrol both below her and above. You want this moment to be tense. You want to play it up, and have it be more than just a single die roll. So what do you do?
Treat it like a fight, so to speak. Or at least like a series of traps.
Break the mountain down into several encounters of particularly troubling parts. A place where Sarah will have to climb with just hands and hang from under an outcropping. A place where she has to jump. A place where she'll be reliant on ledges only about the width of half her finger tip. A place where she'll have to move under a waterfall.
You describe these things to Sarah. She chooses her approach. And then you make a roll. You're not rolling for success failure here though. Well you are, but only partially. You're rolling like it was an attack.
Succeed at the roll, and you do "damage" (or progress) to that part of the mountain. Earn enough and you proceed to the next part. Don't proceed in one roll, and you take damage in return. Maybe not for as long as you normally would if in combat, but your HP goes down.
Climbing the mountain is now an encounter using multiple skills, and at the top of the mountain Sarah is going to be lower on HP than when she started. Or, she can save some HP but make noise (if you want.)
What If It Goes Bad?
Making more rolls, doing damage to the player, and requiring the mountain to be overtaken with rolls increases the risk of things going bad. So what happens if it goes bad? Well, people can and do die or get hurt rock climbing. Not everything works out in the players favor. How it works is up to you.
Someone burning through their stamina climbing a mountain, or escaping a burning building, isn't likely going to be able to fight. Indiana Jones starts Raiders of the Lost Ark in this exact situation. He escapes, he's exhausted, next thing he knows he's surrounded by enemies who are taking his hard earned/stolen idol.
Alternatives to HP
The fun thing here is you don't have to use HP if you don't want to. You can make an arbitrary tract for stamina (or even just a second HP track. Some games have a stun track as is), or you can make an arbitrary tract for whatever it is you want to represent.
Sarah's mountain climb? Instead of HP it could be "Guard Attention." If she takes X points of 'damage' she gets noticed by the guards. Now the tension isn't her becoming exhausted, but her getting noticed while climbing.
Time is another great one. Maybe the player needs to have a combined roll of 85 in 4 rolls in order to succeed. This simplifies it down to basically an extended check, but if you break the task down right you can end up with the character needing to find ways to approach different aspects of the challenge. Now the player has to factor in where they can get the most time saved in order to make up for an area where they're weak.
What If I Don't Have A Mountain?
I used mountain climbing as an example, but you can do it with anything. Any sequence where the characters are doing something under duress can be made more tense with focus. The key is to figure out what the thing the PC is trying to avoid is (in combat you try to avoid running out of HP) and what they're trying to accomplish (combat you try to accomplish winning the fight.) You then find a way, like hit points, to represent that in abstract and give it some focus. Break down the event into a couple choice moments, and present them. Then you just have to wait and see what the player does.
Does this need to happen everytime? Not at all. If Sarah scaling that mountain is just dressing for a larger plan the PCs have that you want to get to, just resolve it with one roll and be done with it (or no rolls!) But if you want it to be more dramatic, have more tension? You just gotta give it the same attention most systems give combat.