Well, it really, really helps to know what other classes and jobs are capable of when you are teaming up with them. It doesn't require playing every job up to 50, but I do think it is reasonable for a player to know what their party members can do.
For example, it really helps to know at what level different healers gain access to revival and status clearing spells. This differs significantly between White Mages and Scholars. So there are dungeons where a White Mage can clear status ailments, when a scholar can't. If you are tanking, this is pretty important information. It is the difference between being able to rely on the healer clearing that poison off of you, and knowing that you need to use an antidote to clear it.
Likewise, it is critical for a tank to be at least generally familiar with the positioning requirements of the Lancer and Pugilist, so that said tank can accommodate them.
It is really easy to take a class for granted if you haven't played it yourself. While you can look up info online about how a class operates, the nitty-gritty of the details can be easily overlooked. For example, Raise doesn't have a recast time, just a long charge time. Reading this, it might seem like White Mages can freely raise dead party members. But that isn't accurate, because MP management is an extremely delicate balance for White Mages. If a White Mage uses up too much mp on Raise, their MP will drop to nothing and the party's healing will dry up. So it becomes unreasonable for someone to expect a raise if deaths are common during a long boss fight. While the coarse details of the spell are easy to look up, the actual implications are something that can only really be known by someone who has spent the time practicing the class.
This isn't an argument that everyone should play every class. Instead, it is a recommendation that everyone actually seek out knowledge about the classes they don't know about from people who have played them extensively. For my part, most of my knowledge of the White Mage is second-hand: my brother plays one. But from his rants, I've learned a lot more about the class than what I could have learned from just reading a skill-list online.
In a lot of cases, the places where this knowledge is important is in the heart of a difficult battles, when you don't have the luxury of taking time to ask other people questions. Part of knowing how to handle any fight is knowing what tools your party has for that fight. A team of Paladin, White Mage, Lancer, and Thaumaturge is very different from Warrior, Scholar, Bard, and Thaumaturge in terms of what kinds of strategy they excel at. Stuff such as AoE capability, Crowd Control capability, support powers, and so on are very different between those two teams. This can and should be a factor in knowing what strategy to use in a dungeon.