The dedicated Healer class needs to be removed.
The Tank needs to be rebalanced.
Problem: A thankless role that is tense, and heaped with responsibility.
Problem: A role that is only valid in PvE and has few PvP applications.
Solution: Remove the healer class. Rebalance the tank class.
For example, in WoW it would be done as follows.
1. Remove Priest class
2. Give every class a small, channel heal that can be applied to allies. Sort of like a bandage, but in combat.
3. Give every class a small, burst heal (perhaps 15-20% health) that can only be applied to themselves on a 2-min cooldown.
4. Make the tank deal lots of damage, but incredibly slowly.
5. Redesign game with above changes in mind.
Now, this may just be me, but I think the only class that is actually fun, is a DPS class. There is not much responsibility on them in raids other than "pew pew pew!" It is the most rewarding, most fun role to play.
I understand that there are people that enjoy exclusively healing, and they still can! They could simply channel heals constantly if they wish.
With these changes, everyone will be a DPS class. Some will deal little damage very quickly. Some will deal a TON of damage very slowly. Some will deal a lot of AoE damage. Some will deal a lot of DoT damage. But everyone will be capable of contributing a significant amount.
I apologize for using WoW terms as an example, but I haven't played many others, and most people here can relate to the examples. So...
Take the Hunter, the Rogue, and the Warlock. Probably the most popular classes in the game, and they are all DPS at heart. Even though they all deal some significant damage, they all have a unique way of doing so. They have the easiest transition between PvP and PvE, because their roles stay the same.
Now imagine 6 classes of DPS. All unique, all balanced, and all fully capable in either PvP or PvE. They all have specialties, but they are all valid in any situation.
This is how we should bridge the growing gap between PvP and PvE. We should stop putting in classes that are designed only for one or the other.