I'm not against regenerating health per se. For genres like platformers and puzzle games, combat isn't in itself the focus of the game. For games like Portal and Mirror's Edge, management resources is not part of the game. They are both inherently obstacle courses that you try to progress through as fast and efficient as possible. A static health system doesn't benefit this in the slightest, it would make gameplay slow and punish the player for experimenting in different ways to navigate the stage. For the former, health is merely a way to make turrets not a one hit kill. For the later, it's a viable punishment for brute forcing your way through the stage in the most linear way possible (i.e. walking through the armed police detachment or simply jumping down that large fall as opposed to finding an easier route down).
For combat oriented game, however, I find that regenerating health is somewhat detrimental to game play. Eschewing the whole "It makes the game way to easy" argument, regenerating health turns every combat section in the game into a series of isolated skirmishes in no way related to each other. Besides a dwindling ammo counter, there is no carry over from the last fight, no real consequences for the choices that you made. This may just be my own tastes speaking for itself, but I like to feel that my choices throughout a game are cumulative. I like the resource management of combat oriented games, I like the notion of a risk/reward system that rewards my for being proactive in a game and not merely kicking down the big bad's door and spraying every inch of his house in lead until I am either out of ammo or dead. For every argument I hear that the speaker is tired of saving at a checkpoint with 1% health and no health packs in sight, I always have to ask "Do you think that has anything to do with what you did before that checkpoint?" I'll admit, there are situations beyond your control that inevitably screws over your survival prospects through no fault of your own, but in most cases I see it's usually because either the player shot themselves in the foot or let the enemy do it for them. For every player I've seen that get caught in the path of a random missile or explosion, I've seen numerous more fire rockets at near point blank range at canon fodder enemies or try to find out whether or not they could kill every enemy in the area with melee only attacks. If you succeed then more power to you, but whatever damage you take through that is just a consequence of the path you chose to take.
In games like the single player campaign of CoD and Battlefield, all regenerating health amounts to is me proning behind cover as I peek up every now and then to shoot a baddie or two before traipsing back into safety until I wipe the raspberry jam off my face. It's no better than dodging bullets while scavenging for med packs around the battlefield during the firefight. It's worse in fact, because while the later rewards my exploration of the map and careful maneuvering, the former merely amounts to you staying in place while the enemy stays in place, waiting for the inevitable moment where one of you gets tired of all the waiting and jumps out of cover to die in a blaze of glory. You're not rewarded for your progress or proactivity, you're rewarded for being stagnant and staying put.
Now, that's not to say that I'm totally against regenerating health. When the concept is well implemented in the game, I actually applaud it. In the Saints Row games, for example, the focus of the game is a cathartic power fantasy trip where violence for violence's own sake is the underlying rule. Regenerating health works well in this, as it reinforces that. In the early Mass Effect games, regenerating health was a skill you could improve on or equipment you could select. That's fine too. I'm actually quite fond of the concept of "partial health regeneration" implemented in games like Resistance, Medal of Honor: Airborne, and some Halo games, where you can regenerate health up to a certain point depending on how much damage you've taken previously. It combines the resource/risk management game play I like, but doesn't overly punish the player for experimenting with combat and always leaves enough room for them to have a chance in the next fight. This is a concept I felt that Halo: CE nailed perfectly with it's implementation of the regenerating over shield, because while there were tangible risks associated with taking too much damage (health loss), the game gave you leeway to dart from cover to cover without the threat of overpoweringly permanent damage (within reason of course) to keep you rooted in place and unwilling to move.
Edit: Going back to the second to the last paragraph about combat in regenerating health games, another aspect I've grown to dislike is the effect it's had on level design. Whereas resource managing games tend to have appropriately large maps to reward progression around the battlefield and exploration, I find that maps of many regenerating health games tend to be very linear and corridor like in design. While some might call this "streamlining", I prefer to call a spade a spade when I see one. More often than not this is probably implemented with the stay prone-behind-a-wall game play in mind, purposefully linearizing the level so that the enemy AI has a hard time flanking the player and flushing them out of cover while they're regenerating health. Whether or not this was implemented to accommodate the player or because programming complex AIs that do more than sit behind cover and occasionally peek out is expensive remains to be seen.
