Note: This was written from the perspective of a World of Warcraft raider, but some of it will apply to other MMOs as well.
Currently, Warcraft has three roles for players to fill: tank, damage dealer, or healer. The first two are distinct from the third, and will be unless major game mechanics are changed. I'm going to set aside encounter-specific mechanics, because I believe they tend to be a wash — for every boss where the tank has something special to do, there's another where the DPS or healers have to be on their toes.
Essentially, a raid consists of everyone racing against one or more risks. A raid boss with no special mechanics, where healer mana regeneration is adequate to handle the incoming damage, is no threat at all — it's ten people pushing buttons for five minutes. But most raid bosses aren't like that.
Tanks are racing against two things: incoming damage from the boss (which they overcome with defensive abilities) and outgoing threat from the damage dealers and healers (which they overcome with threat-building abilities). Tanks even have a neat visual tool for watching their race; it's called Omen.
Currently threat is a non-issue except for uncontrolled mobs wandering around the battlefield; this leaves DTPS (on themselves) vs. mitigation as the tanking challenge. A key point here: most tank cooldowns front-load mitigation ("reduce incoming damage by X% for the next Y seconds"), and tanks who blow their CDs early can fall back on the healers.
Damage dealers race against two things: the healers' mana pool, and special encounter mechanics such as boss enrages/frenzies or adds which do bad things after some amount of time. Not every boss has a doomsday clock behind them, and not every boss that does has a doomsday clock that's meaningful in the face of geared-for-the-tier DPS. This leaves healer mana as the DPS challenge.
Healers continue to race against two things: DTPS (for everyone, not just tanks), and mana regeneration. DTPS goes steadily down as raiders grow familiar with the mechanics, and as they gear up. Mana regeneration only goes up with gear (encounter-specific mechanics aside). Tank healers can fall back on their tanks to use CDs if they get behind, but raid healers are in a tighter spot. But for the most part, healers don't get to front-load their counter to DTPS. The exception is specs with an absorb mechanic, like disc priests, and procs or cooldowns like Guardian Spirit and Ancestral Healing.
Tanks and DPS are racing againt some aspect of the raid. What separates them from healers is that they are capable of getting ahead in that race.
What happens if the tanks lose their race? They die, followed shortly by the rest of the raid. What happens if they win? They takes less and less damage (better armor, smarter uses of their mitigation moves), and can focus more and more on threat-building and DPS moves. But that focus on threat still allows them to contribute meaningfully in their primary role, because it means the DPS aren't threat-capped.
What happens if the DPS lose? The boss enrages and everyone wipes, probably. The healers go OOM. If the DPS win, the encounter just finishes more quickly and the healers are under less strain.
Okay. So what about the healers?
Good healers can proactively throw a heal against damage they know is coming. That said, healers can't get ahead by doing more of what they do (healing) — because heals over a target's maximum HP are lost. Once you've got the raid up enough, you have to stop.
Okay, but what else can they do but push their healing buttons? Their only alternative is to stop healing and use active regen mechanics, or contribute to DPS. Shamans, for example, can use the Telluric Currents glyph to regenerate mana via Lightning Bolt. This used to be a pretty good mechanic until the devs noticed and nerfed it.
Because the developer vision for challenging healers is "make mana management matter", and because overhealing is penalized, healers must be boxed in at both ends — too little throughput and everyone dies, but too much throughput is useless. Healers run squarely into a wall once HPS catches up with DTPS. There is no winning the race.
Thus, healers are being penalized for being too good at their job. Like Cal Naughton, they're forced by their role to ride just behind someone else.
Healers are the only role where the developers' answer for the problem is "step outside your role and do something else". Unfortunately, without the requisite stats (hit, and now expertise), they'll be bad at it. In what scenario will this meaningfully contribute? Tight progression attempts presumably will not have a lot of healer downtime — enough for their damage to make a measurable contribution — and farm content doesn't need that extra damage. It just makes fights go faster, it doesn't contribute to success or failure directly.