I am planning a new campaign setting to our new campaign and because most of it will take place in a Feywildesque other plane (with almost exclusively chaotic settlements) the use of money will be very rare. I was wondering how this affects game balance and if I have to take special care with certain classes or character options.
In Pathfinder, wealth means magic items, and magic items are directly tied to character power and therefore to game balance. The game’s materials assume a certain amount of magical gear for characters, and characters will perform less well than expected without such gear. This makes GMing tools like CR less reliable, because they are assuming magical items that are unavailable.
Worse, these issues are not equally problematic for all characters. Some characters have plenty of their own magic, so even though they’d certainly like some magical items, if they don’t have them they can make do. Others have no magic, and absolutely need magical items because magic is necessary to survive in Pathfinder.
Worst of all, the classes that suffer the most without magical items—non-magical warrior classes—are already the classes that have the most difficulty in the game. So by taking them away, you make a bad problem worse.
For more on how lack of magical item effects can cause issues with CR, the primary GMing tool available, see this question.
But notice that all the issues stem from lack of magic items—not intrisically from lack of wealth, coinage, and so on. Characters need the items—they don’t really need the coin. In fact, coins themselves are useless; it is only once characters can trade those coins for magical items that they actually become relevant to game balance. A character with all his or her wealth in coins and gems is just as badly-off as a character with no wealth at all, as far as the game’s mechanics are concerned.
And, furthermore, it’s not really important that magical items be, well, items. They can be boons or blessings or even (with some flavorful drawbacks) curses. They can even just be an intrinsic part of being higher level: you could say that instead of everyone buying a cloak of resistance +1 at 3rd level, they just get a +1 resistance bonus on all saving throws at 3rd level (and yes, pretty much everyone should buy a cloak of resistance +1 at 3rd level; other bonuses aren’t quite so neat and tidy, but you get the idea).
So what I strongly recommend is that you maintain normal Pathfinder levels of magical item bonuses and benefits, you just change how they are distributed, exchanged, and used to match your campaign setting. This will give you the best of both worlds: the campaign you want, and the most reliable GMing tools that Pathfinder has to offer. (Though, to be honest, even at their best they are still pretty unreliable, unfortunately. It’s just a whole lot worse if you go wildly changing access to magical item effects.)
Just because you as players are using money on paper does not mean the characters are using money. The money is just telling what things are worth. You just as easily be trading via barter or favors, or more realistically debt.
For barter, it can be as simple as I want a fishing net (5gp) I offer this mace(5gp) I found for it.
More realistically such peoples would trade in favors and debt. Bob doctored(1gp) me after the unicorn kicked me in the teeth so I owe Bob about a 1gp worth of something, Steve owes me some cheese (representing 1gp in my "wallet")so I tell him to give it to Bob instead. In this way gp is just an easy way of representing a a more abstract and complicating system. Coins really just represent transferable debt. I helped Dave the shepard so he owes me 1gp worth of wool which I will get in the spring when he shears his sheep, until then it is represented by 1gp in "coin" likewise dave can buy a new set of shears and owe the smith 2gp worth of value until he has something the smith wants.
For the fey it could just as easily be 2gp worth of good dreams for granddad's dagger worth 2gp. Again the gp is just providing a way of quantifying the abstraction. To DM this you just need to make it a bit more work for the players to find a buyer/seller since the players have to search around for someone who wants what they have. You may need to come up with a value for some abstract things like dreams (and a way to actually transfer them).the valves for services and spellcasting will be particularly helpful for this.
You may want to also use a percentage roll to see if they can find someone who wants what they have per hour of searching. I have used this and it can feel quite real if you roll first then narrate it. You can even give a local knowledge bonus to the roll since they would know who likes/needs what.
I use something similar when running games in pre-money/hunter-gatherer civilizations, you search around and trade favors and debt, but you represent it on paper with coin and percentage rolls. I use 10% for really hard to sell stuff(like a book or religious item or in your setting actual gold coins), 30% for specialized stuff and most gear (weapons, fishing poles, raw materials), 50% for common easy to trade stuff most people may use (like clothes, perfume, or food).
This is a very broad question, so I'll answer broadly.
Yes, there are certain classes and character options that will be more affected by a relative lack of money (or, at least, a good place to spend it) than others.
A Sorcerer doesn't need any money for their shtick - they get Eschew Materials for free, so they can throw spells around all day long (limited to spell slots, obviously, and they'd probably tend towards spells without costly material components) in their birthday suit.
Broadly speaking, casters tend to require less in the way of gear to be useful, and their class abilities tend to scale better with level in the absence of wealth. Non-casters tend to be more gear-dependent, and their class abilities tend to scale less well with level in the absence of wealth (eg., a Gunslinger's class abilities focus on getting better with guns, but they largely still require having both a functioning firearm and plenty of ammo for it; a Sorcerer gets more and better spells, and doesn't need anything but their inborn abilities to cast them).