Most MMOs on the market follow the formula of the "Holy Three": the tank, the DPS, and the healer. For example, in a fantasy MMO one would have the knight/paladin in heavy armour up front in close combat with the enemy, a wizard dropping powerful spells to greatly hurt the target, and a cleric casting spells to keep the tank or wizard alive as needed.
Many games gravitate to this model because its "economically optimal", i.e. it makes sense for the character best at healing to focus on healing skills instead of DPS or tanking skills. In fact, in many games a character is locked into a role so firmly that trying to do either of the other rolls is not only sub-optimal, its terribly inefficient.
Eve somewhat follows the pattern. It allows players to concentrate on tanking skills and modules, others for doing as much damage as possible, and there are a couple ship classes dedicated to "healing" in the form of remote armour and sheild repair, as well as energy transfers. But they tend to be very very rare. Why?
The main reason is that any pilot can learn to tank, do DPS, and heal with equal efficiency. There are no classes nor limitations on what can be used so a player does not have to make a decision in the beginning of his character and be stuck with it for the rest of his career. Since healing tends to be less exciting than causing damage, and players can find a balance between tanking and damage, healing tends to be a second or third choice in role in a typical operation.
Secondly, NPCs are completely outclassed by the players in all but rare instances. In level four or less missions, ratting, and many complexes a single experience player in the right ship and setup can tank the incoming damage by themselves so who needs a healer? (This point goes hand in hand with my desire for missions and ratting to be more difficult BTW.)
Thirdly, in PvP the enemy will quickly identify healers and dispatch them first ignoring any obvious tank until later. The Logistic class of cruisers, while being T2, are still just cruisers. Only the heavily armoured Carriers are capable of being effective healers in engagements without threat of quick destruction, and most pilots reserve that kind of tonnage and effort for larger combat situations. Regular small engagements are not ideal carrier deployment areas.
Fourthly, spider-tanking. Related to the first point, since all pilots can do all things, sometimes they try to do so in a cooperative effort in which every ship has a remote repairing module. Then a gang of say 8 battleships can engage and turn the remote reppers on the ship called primary and keep it alive while their combined firepower take out the enemy one at a time. Since all of the ships in the gang are DPS, tank, and healer, there is no weak spot for the enemy to concentrate on. (Of course, there are tactics to use against this one but that is not the purpose of this article).
Finally one more point: counter-intuitively, the death penalty encourages less healing. How you ask? Since the ship and mods are lost upon destruction, this leads pilots to often fly less than perfect ships and modules for their skills (i.e. don't fly what you can't afford to lose). Subsequently, we get into the mindset of expecting to lose our ships at any time during an operation and knowing that the loss of one ship does not affect the whole too drastically in many cases, hence less need for a healer to keep us alive.
To summarize, Eve rewards balance and flexibility over specialization most of the time.