Q:

What kinds of conflict-resolution can be fun beyond combat in an RPG?

As someone who grew up on the single-digit Final Fantasy games and AD&D, there are few ways of solving an rpg problem as fun as a fight. Different streams of accomplishment (object acquisition/crafting, skill trees, puzzles, player strategies, team combination/synergies, etc) all go into a few minutes of frenetic action, allowing for the feeling of accomplishment and progress via stat increases as the game progresses. 
That said, we tend to characterize rpg players as murderhobos for good reason. When you incentivize killing things, it's not long before all problems start looking like something to hit with your sword. And when you're able to steal anything not nailed down, why not do it?
While other kinds of conflict-resolution can be fun when in their element (farm/ Civ/Sim games for the sense of building something, spy/ninja/assassin games for deftness and exploiting situations, social games for roleplaying a character or out-thinking the other guy), they're often not fun outside their very narrow venue. And because we're limited to conversation with a computer (at least in single-player games), social games can't even explore the whole of their venue. 
I'm racking my brain trying to figure out how to properly gamify the societal ideals of don't-kill, don't-steal in such a way that attacking or stealing becomes a conscious choice, not just a default means of action...but at the same time keep the same diversity of fun-inputs that fighting/stealing games have. Most games just code in consequences: the town guards are sent after you and/or some faction is less willing to deal with you. But this feels unnatural, and only invites further gaming of the system, producing better murderhobos at the end.   
My reason for doing so? The main theme of my game is a heavily philosophical one(think a halfway-understandable Xenosaga) on how morality might play out in the post-post apocalypse. While killing and stealing would be a part of that, I feel that our natural tendency as gamers wouldn't allow this question to play out 'fairly'. 
Any idea on how to put killing and stealing on a truly equal playing field with other forms of conflict resolution?

A:

Why don't we constantly steal and murder in the real world?

We have empathy with other people and don't want them to feel bad.
We are afraid of getting caught and punished.

Why do games often fail to convey these hinderances to committing crimes? 
Creating empathy
There are actually quite a lot of games which manage to create empathy for NPCs, but usually only for a sub-class of the NPCs in the game. Most games clearly separate the world population into those NPCs you can murder and steal from and those you can not. 
In the Elder Scrolls series, for example, the player will get punished if they commit crimes against NPCs in a city (and get caught). This punishment might be a game mechanic (like guards hunting down the player) or a meta-mechanic (quests become unavailable when the player kills key NPCs). So most players will only do this when they want to experience the fantasy of playing an immoral thief and/or assassin (which isn't wrong per-se as long as the game handles this well). 
But there are no repercussions whatsoever for going to a bandit camp, killing everyone there and stealing all their possessions. Even players trying to roleplay a "good" character will do this, because the game communicates to them that this is something they are supposed to do. Is this a moral thing to do? No, most societies would prefer to catch those bandits alive and put them on trial for their crimes. And even if they approve this kind of vigilante justice, they would at least expect that the things they have stolen get returned to the rightful owners. But most games do not care about this. Bandits are just moving targets to hit and their loot is the reward for doing this well.
If you want to avoid this, do not have such a class hierarchy of humanized NPCs and dehumanized NPCs. Make sure that criminals are people too. Give them names, backstories, friends and families. Also give the player some other meaningful ways to interact with them besides fighting them.
Fear of punishment
Your conscience is the voice in your head constantly whispering to you that someone might find out what you have done.
In order to invoke this feeling in your game, you need to make sure that committing crimes against people is handled in a plausible way. Many games will give you an immediate feedback if you get caught. But that's not how crime works in the real world. If I steal something from a friend's house, they might not catch me immediately. But over time they might notice that their stuff goes missing everytime I was at their place. They might call me out on this, but I will deny. Then they see their stuff in my possession and won't want to be my friends anymore. Then they murder me. It might take years until the police finds my body and locks up the murderers. This delayed form of punishment is one important reason why we are afraid of committing crimes in the real world.
Also remember the last section: Criminals are people, too. When you harm a criminal, then lawful people might forgive you or even approve. But criminals also have friends and families... and mob bosses who dislike people who interfere with their business. The punishment for harming a criminal might be even harsher than anything you might expect from a court of law.
Most games do not simulate this because it can become quite complex. 
One problem is that it is very difficult to communicate to the player which acts exactly lead to bad consequences at a later point in the game. When a game punishes the player but they have no idea what they did wrong, then the player can't learn from this experience. So you should invest quite a lot of resources into communicating this well. One way I could think of would be to replay a flashback of the player committing the crime to remind them of what exactly they did wrong.
This whole mechanic goes against the usual expectations of the player. So you need to teach the player early that their actions have delayed consequences. Tempting the player to commit a petty crime and then calling them out for it a few minutes later should be part of your tutorial.
Another problem is that when you allow the player to antagonize every single NPC in the game, then they might break the plot of your game. This is why I would rather recommend to experiment with this in a game where the plot isn't pre-written but rather created through emergent gameplay.

A:

The core of most RPGs tend to be around their combat system, items increase your ability to fight, progress depends on winning fight, classes and levels focus on combat ability.
But the mechanics of turns and use of items and abilities don't need to be only useful in a system to represent an abstraction of combat, similar mechanics could be used to apply to dealing with characters in other ways,
Rather then dialog trees, use the mechanics of combat in RPGs.
Example :
A guard blocks the entrance to the castle, the party engages the guard, in conversation, he has high resistance to intimidation and medium resistance to persuasion and deception
The party members can have turns to use abilities that can affect different 'health' bars representing how close they are to being persuaded/intimidated/deceived/cheered up;
Abilities could be depend on non combat classes a [Deceptive] character could use 'Confusing' to do deceptive damage and make the guard miss have next turn encourage your party rather then its intended result.
The guard could have 'Threaten' which intimidates your party, and could result in losing the encounter. Special abilities could include calling in more guards.
There could be healers ('Reassure') buffers, and equipment and stats that buff your characters, a large strong character might have high intimidate, and stacking it with gear like could be used to increase. (Could be easily confused by deceptive characters)
Collecting items, giving gifts that have high persuasion (or 'knowledge items' that work on specific characters') like finding out a characters secret and using it in a fight as +X intimidate).. or using gold to bribe
All of this would allow non combat situations to have the addictive elements of an RPG (leveling, progression, strategy, collecting)
Also unlike combat there is a bit more room for different outcomes,
If you're party loses to a guard by intimidating it could lead to them being imprisoned and having to escape;
If they intimidate the guard they might be able to get in the main gate, but have to deal with different encounters then a persuasion might (an escort to another npc) and the story branches are determined by mechanical results

A:

Rewards
One approach might be to make the NPCs themselves, and their support, resources worth acquiring. Design the incentives and core resources around things that only other characters in the game world can provide. Focus on mechanics that require the NPC to remain alive and friendly towards the player.

Perhaps the player has an agenda they need to push, a religion or philosophy they promote, or they're just looking to become famous. Every friend or follower gained increases their influence, which is tied to the main objective(s) and/or character progression.
If the game allows for it, let most NPCs be recruited as followers, temporarily or permanently. Balance the difficulty around the player having followers.
Let friendly NPCs provide resources or items that are otherwise scarce, and need to be "refilled". Depending on your settings that might be necessities like food and shelter, healing potions, ammunition etc. Make reputation a resource the player can spend on this or resrict access to a positive reputation (as opposed to "they didn't hurt me yet so I guess they're alright.")
Tie experience or skill upgrades to bits of information obtained (primarily) from NPCs. Maybe a lot of knowledge has been lost in the apocalypse and most people only know fragments. For example, each NPC could have 2-3 skills they could increase by one point, or a single magic spell they know. If you want to master a skill, you'll have to persuade a lot of people to teach you.

Creative interaction
Some open world RPGs allow you to invest in NPCs to improve their services or make them stronger followers, but that's usually reserved for e.g. store owners or designated party members. You could allow the player to shape most NPCs or factions by "giving back". Allow them to build or name things, give equipment to NPCs, claim areas for them. Most players will care more about a town they helped design than one they're just passing through, and later on, when the villain convinces everyone they're a criminal, they might be more inclined to persuade the guards to let them speak to the council and prove their innocence than to burn it all to the ground.
It's not necessarily either-or
Many games allow for social encounters to progress into combat ("The bandits seem unimpressed, they draw their weapons") but not the other way around. Allow the bandit leader to negotiate their surrender after the player kicked a few asses, and give a substantial reward for capturing criminals. Give the player the same option if they end up in a tough spot. Make the difficulty of a fight something that's hard to assess at a glance, but give hints in conversations.
Tweaks for "social combat" systems
Physical combat in most games has a few aspects, for example terrain and positioning, that are easy to intuitively understand and can be used without much effort to add depth to a combat system. These aspects are lost if you just copy and "reskin" the same system for social encounters, and technology isn't yet at the level where we can randomly generate compelling dialog (while generating a tilemap is easy). Most (video) games that use this kind of mechanic handle it in a very abstract and often repetitive way.
I'd recommend looking at card games for elements that add a degree of guessing and puzzle solving to your system, or even base the system entirely on a card game (with or without actually rendering/refering to any cards).
Classical card games combine a tweakable randomness with the ability to "out-think" your opponent and can be complex enough to be engaging to play against an AI.
TCG style systems allow the player to collect "cards" (secrets, skills, information, a specific reputation) and use them in combination for synergy effects. Perhaps executing the thief, bringing her to the authorities or letting her go will each give you a different, unique card that can be played later when you encounter the thieves' guild?

