No.

You probably have opportunities in life to enrich yourself unjustly. And you probably don't. Character A is no different from you.

But it sounds like you have a problem, not just a question:

It sounds like the two factions of players are playing two different games.

I won't even label the two games, nor characterize them. Suffice it to say that at a table with 4 players and a GM, there are likely a dozen different, layered games being played simultaneously. Some of them have to do with the characters, even!

The proper meta-action to handle this is not for one player to mock or pressure another, nor for one to unilaterally put the brakes on the rest. The proper way to handle this is to acknowledge it, have a brief conversation about what each player enjoys and doesn't, and to move on with the story. No one's playstyle need change, but each player should respect each others' desires. So no one can force the Paladin to commit robbery but the Paladin can't stop the others, either.

If that means character A helps get a kitten out of a tree while the other three go a-robbing, so be it. That's a much-easier spotlight split to handle than when they get separated in a dungeon!

Constant party splitting isn't great long-term, but the long-term is an in-game problem that the players need to recognize and then tackle. Your problem is the RL table-dynamic. Point it out and expect mutual respect.

Generally... no. You shouldn't need to give up your character's integrity. There's usually alternatives and compromises available.

Now, there is My Guy Syndrome, which is that you'll do unfun things because "it's what my guy would do", without realising you have a choice there. There's also Making the Tough Decisions which is "you can shape your character to help propel the plot". This is neither of those, this is players pressuring Player A to drop what they feel is a core part of their character and source of fun.

For some people, those core parts are important. Some players derive their joy in a game from playing out a character — it's a totally valid mode of play. They should still avoid My Guy & make tough decisions, it's not mutually exclusive. Those players are doing something unwelcome, and pressuring a fellow player to give up their source of fun and compromise their values. It's not healthy play.

So: those alternatives and compromises are how to move things forward. These players, A and B and all the others, need to sit down and talk about third possibilities beside simply "everyone including the paladin commits robbery" or "no robbery at all". There's win-win alternatives to be had if you start talking them through seriously.

Situations like the one you describe can actually create some beautiful drama to explore. There's How do I play a paladin without being a stick in the mud? to provide some basics for figuring out how to play a super-lawful paladin alongside less lawful activities, but one of the things that paladin can do is simply go along with his friends to keep them safe. They're stupid, they're doing something stupid, he hopes the robbery fails and won't steal anything himself, but he'll at least be there while they're doing it because he cares about his fellow party members and doesn't want them killed. (Does he not care about them? That might be one of those "make a tough decision" things: time to develop that the paladin kinda grudgingly cares about them, or something.)

Alternately, the paladin can at least keep watch outside for more or less the same reasons, if splitting the party is workable. He could be there to prevent the absolute worst, or stay aside to be able to vouch for them to authorities if they're caught and otherwise get them out of a mess (as RollingFeles suggests).

The paladin might not be totally happy with it, but that is a drama seed you can use for fun and character development. People get forced into unpleasant situations sometimes, or give up something for the sake of others' benefit.

(All of that considered, Player A should probably not force the party to do things their own character would not themselves do, since that would be the typical stick-in-the-mud paladin. It doesn't sound like that's happening, though.)

In summary

If Player A would feel it compromising and unwelcome to have his paladin participate in the robbery, his fellow players should respect that. Rather than asking him to quit fussing about it and go do some robbery, they should take that constraint, accept it, and discuss alternate possibilities. Player A should work out something with them that doesn't violate their core fun values, and look at compromising situations as a potential opportunity for character development and drama and its own kind of fun — but Players B onwards should respect their boundaries.

Is it necessary? No. But nor is it necessarily optional. On the one hand, My Guy Syndrome is a real thing; It's entirely possible to play a character in a way that makes the game worse for everyone. On the other hand, forcing a player to play their character in a way that they don't enjoy is also a real thing, and hurts the enjoyment of that player - and, if others in the group care about verisimilitude, potentially their enjoyment, as well. In other words, blindly calling "My Guy Syndrome" every time a player has in-character reservations about something is bad for exactly the same reason that My Guy Syndrome can be bad: Someone is following an arbitrary rule of behaviour even when it makes things less fun.

The trick to making sure everyone has fun isn't to blanket ban a certain playstyle, nor to always allow it, but to find a comfortable balance: A balance between the different activities each player finds fun, a balance between how much in-character decision-making is too much or too little, a balance between how much time is spent on table talk and how much is spent in-character, and so on. There's no One True Way to play a game, and every group, GM, and player has different preferences, so every group will strike that balance in a slightly different place, and the only way to find out what works best for your group is to talk it over and experiment.

So, begin a discussion! Suggest the same-page tool (NOTE: It's supposed to be a list of discussion points, not a survey) or some other way of investigating what it is the group does and doesn't want to do. Presumably the group wants some in-character decision making, since that's the very definition of role-playing; Try and narrow down how much of that is too much for the other players, and how much of it is too little for you.

Hopefully, you'll find some overlap - and then you'll find a balance that works for everyone. If not... Well, that's OK, too. It just means your character - and possibly your playstyle - isn't a great fit for this group; If that seems to be the case you'll be best off leaving and finding a new game.

Oh, and one final thing: Paladins are hard. The class seems to be designed on the assumption that you want to play Sir Lancelot - and Sir Lancelot comes from a very specific kind of story, in which might serves right and the virtue of the hero is never in question until he or she tragically falls from grace through his own failings. From your description, it sounds like the other players want a campaign where they're scoundrels, anti-heroes and ne'er-do-wells wielding sword and spell for fortune and glory. That's an awesome sort of story too, but it's incompatible with the Sir Lancelot one. If you find this kind of problem comes up a lot, maybe retire your Paladin and roll up an honest thief. Assuming, of course, that that appeals to you.

My answer is... maybe.

The question to me is "What game are you playing?" I'm not talking about game system here, either. You are apparently using D&D or Pathfinder in this example. That's irrelevant. There are a large number of ways to play D&D. In some of those games getting more loot IS the point and inter-character conflict is anathema. In some games the development and conflict of character personalities is the game and getting more loot is very secondary. There's a metric crap-ton of other ways to play as well.

So, which game is being played here?

Without sitting at the table, it'd be hard for me to know for sure. However, I think it is telling that 3 players are putting pressure on the 4th player to change what he is doing. It isn't 3 characters are putting in-game pressure on the 4th character. Those are different situations. It seems to me that 3 of the players are saying, "Look, dude. The point of the game is to get phat loot! The whole goodie two-shoes thing is just fluff. Get with the program." That sounds to me like the players are playing the game with differing sets of assumptions about the point of the game. No rules on Alignment or class features are going to fix that.

My advice is this... Figure out which game you're playing and then do what aligns with the expectations for that game. Maybe that means the Paladin character sticks to his guns. Maybe it means he doesn't. Maybe it means he helps but the players construct some in-game way that the Paladin can justify the action.

Anyway. My 2 cents.

No, it should not be necessary for anyone to sacrifice their character integrity for the sake of the game. But note the emphasis and the implications: My phrasing implies that the character integrity is fully formed, or strongly developed, at the time of the sacrifice. You cannot really sacrifice what you do not yet have.

However, it is nearly always necessary for everyone to harmonize their character integrity before the game starts, and generally to keep things in harmony as the game goes one. This might mean no moralizing paladins in a party of morally grey bandits, or, equally, no self-centered thieves in a party of religious crusaders. Or any number of similar scenarios.

Note that ideally, the GM will facilitate this harmonization process.

As a side note, though, subtler conflicts can arise in play-- even in a morally grey campaign, "Is the 16 year old child of the evil king fair game? Does it matter if the 16 year old child has been leading armies?" might generate different answers, and those can be very satisfying to work and play through. But those are usually the result of detailed role-play, not set-up issues like "Looters or crusaders?"

Teamwork Is an Embedded Assumption in most role playing games.

There are two different points that need to be addressed at the table, but it's obvious that team building for these four is incomplete.

Point 1:

"Dude, just go with it this time, forget the roleplaying".

This player needs to be reminded what kind of game is being played: a role playing game. If Player A has been consistently role playing, then it is rude of Player X to demand that Player A stop.

Point 2:

For what purpose have these four gotten together to adventure? If they have not sorted that out within the game world's context, then it seems pointless to continue play until the four of them come up with a reason that they have formed an adventuring band.

For the scenario in question: while splitting the party is likely sub-optimal, if all four agree that three will stage a raid and the fourth won't, then run the robbery with 3 and address with Player A what his activities are in parallel.

Sequel A: are you about to encounter a PvP scenario? If Player A goes and reports the robbery to the local sheriff or town guard, since he's trying to stay in character, the potential for the "my guy" problem to arise has struck. That sequel can wreck party cohesion, both in game and out of game.

Sequel B: robbery successful, next adventure?

Sequel C: robbery unsuccessful, some of the 3 hurt or captured. How does Paladin help out/free his friends? Some rich role play is possible here.

This brings us back to the core question for the four players to answer:

Why are the four of us doing stuff together in the first place? Why are we a team who take on deadly challenges and trust/need one another to keep each other alive while doing so?

Until you can get the players to arrive at the answer to that within the context of your game world, the bickering you are seeing won't go away.

Respect the metagame Essentially the metagame here is that the characters need a reason to stick together because the players have to in order to play the game.

So something has to give in order for this situation to work. Talk it out, the players to have to come to some sort of compromise about having a paladin who is "constantly" at odds with general party behaviour. Either it is resolved with the paladin leaving the party, the party behaving like responsible citizens or maybe the paladin becomes a bit more Arnald Amalric in thier outlook on Lawful Good

The main question is, why are the "heroes" robbing a local noble?

Maybe he is evil, so the paladin can be convinced, not only to rob the noble, but also bring him to justice.

If the noble isn't evil, and the paladin can't be tricked into believing he is evil, the "heroes" have a problem. There is no way the paladin will participate into this act. Most likely, he will be out warning the city guard so the "heroes" can be arrested when they leave the building with their loot. Maybe they serve jail time, or even worse, but that are the risks of the play.

An advise to players everywhere: don't include the paladin into your shady plans. And if he finds out, convince him that it has a noble cause.