There are a lot of games that treat non-combat actions with complexity and sophistication. As mentioned by others, some games have specific subsystems for noncombat conflicts, like Burning Wheel's Duel of Wits: A set of options like Point, Rebuttal, and Obfuscate, with the position you're arguing assigned a hitpoint total.
Other games employ a universal procedure to cover a variety of conflicts and contests. Mouse Guard, for instance, refines Duel of Wits to the point where you have a single set of actions: Attack, defend, Feint, Maneuver, which can be "skinned" fictionally to cover any sort of competition, from a fight to a debate to a chase. Dogs in the Vineyard has a "combat" system that covers Talking, Physical, Fighting and Shooting, with the win/lose stakes defined up-front, anything from "convince him to repent" to "kill him dead." Smallville RPG has a Contest procedure where characters can wrangle over convincing each other to change their behavior, or "convincing" each other to get punched in the face and fall down.
The game The Shadow of Yesterday (and its generic variant, Solar System) has a system where everything, fighting included, is resolved in a single roll--unless someone wants to contest the result, in which case you enter bringing Down the Pain, a series of back-and-forth combat-like rolls that can encompass any sort of action that's applicable to your goal.
A couple of common features to all these rulesets: 1, a focus on only invoking the complex resolution for actions that will be fun to play out in detail. In Dogs if a conflict is unimportant you just get your way; in BW, MG and TSoY you have a single-roll check for most actions. In Smallville you only use Contests in direct conflict with major characters; vs. Mooks or to, say, stop a runaway train, you use a single-roll system. On a related note, these games all depend on taking action that's important to the character's beliefs and goals; if someone wants to do a simple action like hopping a fence, it's either a gimme, or else you ask why the character wants to hop the fence until you arrive at what's really at stake in the scene.
The second common feature is, the results aren't just win/lose: there are additional consequences for engaging in conflicts. In BW's Duel of Wits, you might win the argument but have to agree to a Compromise. In Mouse Guard the GM inflicts Conditions on you--Angry, tired, sick, etc--as a result of Tests and Conflicts. Similarly, in Smallville you inflict Stress on your opponent--Angry, Insecure, Injured, etc. independently of (or instead of) getting what you want. In TSoY Bringing Down the Pain means trading a bunch of harm back and forth, both physical and emotional. And in Dogs, when you Take a Blow (have a low defense roll, essentially) in a conflict, you suffer Fallout--from stinging shaming words, to the bullet that might kill you--whether you win or lose.
This is one of the biggest benefits of fleshing out the noncombat portions of a game design: you can focus on a wider, richer array of fictional consequences for your characters. Instead of simply living or dying, you can see characters shamed, terrified, exhausted, self-doubting, or heartbroken--all above board and by the rules!