For anyone who wants a LARP more complicated than a six-person melodrama, conflict resolution is a big part of the game. In LARPs that go on for more than one session (such as Vampire LARPs or various boff-weapon LARPs) the conflict resolution is the game, and the rest is just a momentary scenario.
In essence, the difficulty is in keeping the resolution simple enough that it can be done without being burdening while still being complex enough to make it thought-provoking.
There are a lot of methods of resolving conflict in a LARP, most of which are transparent. Usually there's some kind of tiered system which, if the two in conflict are close enough, devolves to some kind of chance (typically Rock-Paper-Scissors). Some LARPs use a more subtle system, such as giving each player five cards, and letting them choose which to use up in a conflict...
But those are just simple tools to make the game run smoothly. The real issues in conflict resolution don't lie in how you make your decision, but the results of the decision and how it stretches to include other mechanics.
For example, I ran Zombie: The Brain Eatening. I used a simple rock-paper-scissors mechanic, but unlike virtually every other LARP ever, a tie meant that both parties hit the other. This meant that the game had a lot more dying, especially in team situations, where anyone on the other team that you didn't beat in RPS hit you. In essence, any challenge could result in injury unless you were immune to their level of damage (which wasn't very common).
Similarly, everyone had three states: okay, wounded, and dead. This made tracking death fairly straight-forward. There was no HP counter, no doing X damage of Y type.
I'm not saying this is the way to go. You have to consider the results you're aiming for.
For me, I wanted a horde game with an awful lot of dying in it. So, obviously, I chose a ruleset which resulted in people dying. This ruleset is not what you would want if you were running a LARP with longer-lived characters.
Higher complexity is acceptable, especially in games where every character is intended to be complex. The idea with this is that characters should have longer-term effects from their encounters, but not be out of the game unless something extraordinary happened.
Also, not all conflicts are physical. There are a wide variety of conflict types. You might have one for arguing philosophy, or influencing voters, or changing the past, or sneaking. You can choose to centralize, use the same mechanic for all your conflict types. This will make the game feel very conflict-driven. Alternately, you can decentralize, and use various conflict types.
An example of this would be from a vampire LARP, where obfuscate renders you invisible and auspex lets you see invisible people. Instead of explicitly stating things and maybe rock-paper-scissoring, the obfuscator is careful to show his level in the number of fingers he holds up. The auspexor can simply compare for himself. This is a very passive-aggressive method of conflict, and both sides of the equation will often find themselves screwed over pretty much at random from time to time. Sort of what vampire is known for.
I hate vampire LARPs.
Anyway, the point is that you want your game to have a given feel. The types of conflicts you have and what happens when they are resolved will give your game the feel you want. In addition, it can also provide you with grist for entertaining your players if it is deep enough. If it is shallow, that's fine, but you'll have to find some other method to keep them entertained.
There are plenty of fun LARPs that go both ways.