Active vs. passive resource systems
To really discuss the role of mana, we need to consider how it is currently used. It's fair to say that mana is only directly used in a tanking role by one class because passive resources like mana (ones that are simply there, rather than generated by player action like rage and runic power) aren't terribly compelling for the tanking role. The one class that tanks with mana does so by essentially ignoring it, using Judgements to regain mana in order to use that mana on other abilities. In essence, the paladin tank uses an active resource system. Some abilities cost mana, others regenerate mana, while still others generate holy power that is then spent on still other abilities.
An active resource system (do X to get Y to spend on Z) has positive and negative aspects. Active resource systems are throttled. You can't do your biggest and best move out of the gate; you have to ramp up to it by doing the things that generate the resource. This creates ramp-up time, which can itself be an issue for classes (the perennial roller coaster of retribution paladin DPS in Cataclysm is in part based on this phenomenon), especially when the ramp-up is dependent on keeping uptime on a boss. Just having all your resource up front means you can unload all of it up front, at least compared to a system where you first have to do X before you even have Y to spend on Z. However, the limitation of a passive resource system is that it can run out; if it can't, it's not a resource at all.
Examples of active resource systems in WoW include (but are not limited to) the death knight's runes and runic power, rogue combo points (energy itself is passive, as it simply regenerates over time, although that's an oversimplification for purposes of discussion), warrior and druid rage and hunter focus. Focus actually combines active and passive aspects. It starts at full (like mana and energy) and regenerates slowly over time, but hunters use specific attacks or abilities like Steady or Cobra Shot to increase their regeneration.
Up close vs. far away
For tanks and melee DPS, it seems clear that mana is a non-system. Both protection and retribution paladins have effectively abandoned it with the addition of the holy power system. What mana they have really only serves to prime the pump, so to speak, and gets them to work generating holy power, and they regenerate their mana via judging their targets. Enhancement shaman in melee range use Shamanistic Rage to simply ignore mana costs for 15 seconds out of every minute while Mental Quickness greatly discounts their mana costs and Primal Wisdom regenerates mana constantly. It can be argued that enhancement shaman have simply abandoned resource management entirely.
This does lead us to consider that, for tanking, melee DPS, and at least one ranged DPS type, mana is already dead. Even classes that use mana in the tanking and melee roles either use it as a subsystem for a larger active resource or have made its regeneration so trivial that they don't actually need to manage it in any significant way. Ranged DPS classes use mana more significantly, but even for shadow priests, balance druids, elemental shaman, mages and warlocks, mana regeneration simply isn't the limiting factor in most content. (Moonkin, who have to manage Eclipse states, are fairly close to an active resource system as it is.)
Healers play the key role
This leaves healers, who have the most complicated relationship with mana. Even holy paladins (who have a secondary resource system that pushes them towards an active model) are still at the mercy of mana pool and regeneration. Fight tuning makes use of the limitations of the mana model to provide a way to end fights.
This is a deliberate mechanical choice, and I'd argue it's a good one. Six-hour fights aren't fun. Imagine if your Firelands 10-man could kill Shannox by simply whittling him down over the course of six hours. Would you want to? Other mechanics exist to balance this sort of experience (hard and soft enrages, increasing debuffs that must be juggled), but exhausting mana is a tool in the encounter balance tool kit.
For PvP content, it's even more crucial, because without the ability to exhaust healer mana, PvP can become bogged down. Active resource systems for healers would also require heavy redesign of how healing works. The monk as a healer is going to employ a hybrid system that uses mana to heal but also moves into melee in order to generate Light and Dark Force abilities.
It's certainly possible to imagine all healers moving to a system like holy power, but is it desirable? Going even further and abandoning mana for healers would have to be very heavily balanced and would effectively be a ground-level redesign of the game. If mana were no longer a limitation in the way it is now, new systems would have to be introduced or all bosses would have to have mechanics that would prevent fights from simply becoming grueling battles of attrition. PvP balance in a purely active resource system would need to shift toward a denial-of-resources strategy, with interrupts and stuns/snares taking on even more importance, since a character who can't get their resource gain attacks off can't gather resources.
The final reckoning
In the end, I think we'll see more active resource systems. The description of how warlocks are going to work in Mists of Pandaria definitely makes me think another class is headed toward a hybrid resource system with heavy active elements, and in time, we may see the complete departure from mana for non-healers. But the balance headache -- and more importantly, the loss of mana as a tool in balancing content, should healers abandon it -- makes me think we'll always have mana in one form or another. Active resource generation means hitting buttons to gather your resources, and hitting buttons is always fun. But it probably wouldn't be as fun for people already tasked with the responsibility of keeping everyone else alive, and it would cost a lot to make it viable ... perhaps more than we'd gain.
It could be done, though. The monk will show us how far the active system can be pushed for a healer, ultimately. If it is a wild success, expect mana to undergo serious changes.
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