In almost any pug, 5-man or raid, the first question that will pop up is: Whose fault was it? The healer(s) or the tank?
I've encountered the situation several times. It's just as embarrassing and annoying every time.
The blame game
The main problem is the focus of the group: The focus should not be "whom to blame?", but "what can we do next time to prevent it from happening again?". It's freakin' hard to admit you did any mistakes when you need to eat through a heck of a lot of mean words in addition to the weight your conscience is already putting on your mind. Most people prefer to remain silent in such a situation, and that's a very bad starting point for finding out what could be done differently.
There are times when you just need to replace someone in your pug. However, that ought to be plan B or C, not plan A. Plan A should always be to try again, improving from last time.
The people that like to blame are also prone to leaving whenever something goes wrong. Just yesterday, we did the weekly raid quest to kill Razorscale in Ulduar 25. We waded in with no plan whatsoever and wiped at 2%. People started leaving. What the heck? 2%? It's nothing left. We were practically guaranteed to get him down on next attempt, and we did. Similarly, when the new 5-man instances just were released, I wiped with my group 5 times on Garfrost because people insisted on doing it as a dps race. While that is possible, it requires Frost Resistance Aura, Frost Resistance Totem, a very special setup consisting of only classes capable of removing debuffs including a paladin healer, or lots of lucky resists. Several people came and left. Only me and one other persisted. And we learned the encounter the hard way, how we were supposed to hide behind the boulders to clear the debuff stack. We got him down. It felt just as good as downing any raid boss the first time.
Some people cannot be unbroken
There might be some reasons for replacing people, however. Unwillingness to learn is the primary reason I want to kick people. People insisting on repeating the same mistakes despite a different strategy being agreed on in the chat. I'll give another example.
We wiped times and times on the 9th wave in Halls of Reflection. The healer complained that I was taking too much damage, he couldn't possibly keep up. He wasn't superiorly geared, but neither was he particularly undergeared for the instance. He said I wasn't usually taking too much damage, except when I got stunned. Not strange. I got almost 60% avoidance normally, but when I get stunned I avoid 0% damage. That means I take more than twice the normal damage. We discussed in party chat, and decided on killing the mercenaries first. It worked like a charm. Except for one rogue, who insisted on killing the priests first, because "that's what had worked for him earlier". That meant I had to maintain lots of threat on two targets rather than one, and it took longer to burn down the primary target. Annoying. We were, however, successful. However, I dislike people doing it like that. At the very least, he could've protested on party chat, letting us know he disagreed before we actually got into the fray.
Call me undergeared, huh?
Another thing that annoys me is people entering the group, taking a look at my unbuffed health (about 32k now, was around 30k earlier) and leaving. They consider me undergeared because I don't have 40k health. That buggers me seriously off.
Before my current career as a DK tank, I was a druid tank and healer. Once I was pulled into a group in Trial of the Champion heroic, as healer. The tank was a druid with 45k health self-buffed. Great, I thought, the tank is well equipped, this ought to be easy to heal.
It wasn't. It was actually harder to heal than most other tanks I had healed. Why? It turned out he had skipped on all kinds of avoidance to buff his health sky-high. That meant he took much more damage over the instance than one who had had more balanced stats.
Consider it this way: Let's say the tank is going to take 1 million damage before avoidance but including mitigation (armor). If he avoids 30% of the damage, he will take a total of 700.000 damage. If he avoids 50% of the damage, he will take a total of 500.000 damage. That's 28.5% (the difference, 200.000, divided by the total, 700.000) less damage taken, a huge difference. Note that the difference is more than 20%, the actual difference in avoidance.
I go with balanced stats. That yields me less health than tanks going all-out stamina, but allows me to take less damage over the instance. While it is less effective against magic damage, there are almost no encounter that does not combine magical damage with physical (avoidable) damage. take Koralon in VoA 25. He hits for a lot of unavoidable magical damage with his meteor fist. At the same time, he auto-attacks. If I can parry or dodge on of those, I'll have better chances on getting healed in time compared to a tank who avoids nothing but has more health to soak the damage.
No, you can't depend on avoidance. At the same time, you can. The waves in Halls of Reflection heroic hits for a lot together. Since they hit separately, you get a lot more attacks incoming per second than a single boss with the same total damage would've given you. That's where you can depend on avoidance: As you take more hits, the number of actually avoided attacks get closer and closer to the number on your character sheet. See the difference between how much damage the tank normally and when he's stunned on those encounters, and you'll see.
I do have the Black Heart myself, but I've skipped it in favor of an avoidance and a mitigation trinket of higher item level.
When healing fails
Sometimes, though, it's just not enough. You've got a properly equipped tank, he's not taking unnecessary damage, and he's holding threat on all mobs. Blizz has in the latest 5-mans added a lot of abilities which hits a random target member. You need to keep everyone alive while not losing the tank.
There are a few things that can be done differently. Talents might matter. Sometimes they make a major difference. Once we had trouble with an encounter, I (tank) died first every time. The healer was a bit underequipped, but not horribly much so. He shared his frustration with us. I inspected the druid healer to see that he had skipped Nature's Grace, one of the most hardcore tank-healing talents. However, in most cases, talents are not the problem.
However, priority often seem to be a problem. Many healers overreach themselves. Some need to be told that while healing, the tank and the healer is priority. If they have trouble keeping up, let dps die first, not the tank. In particular, this holds true when dps takes excessive damage. Not moving out of the AoE on the ground? Dpsing while everyone else hides? Let them die. Sometimes the healer can carry them through their mistakes, but that's not how it is supposed to be done. If the healer has trouble keeping everyone up, they should die and usually will die first.
There might be some parts of the encounter that someone in the group is not familiar with. I've beat encounters with seriously underequipped people as long as they know exactly what to do.
Often a dps can do something different that can change the outcome of the fight. CC a mob? Use a different totem (like Grounding Totem or Cleansing Totem)? Change Aura? Stun more, even at the expense of dps? Put a pet on a mob, letting it "tank" it and die rather than letting the tank take more damage?
Finally, most encounters are impossible to do without enough dps. If dps is lacking, it's neither the healer nor the tank's fault.
How to spot "problems" and solve them
I have three primary tools.
First, inspect. It will give you an idea of what gear level the tank or healer is decked in, and what talents he is using. A Blood tank is going to bring a lot more self-healing, but less AoE-threat. A paladin tank will usually take less damage from most hits because they block so much, but have no particularly defense against magic.
Second, recount. It will tell me what kind of spells the healer is using. Is the healer having trouble keeping you up? Perhaps he's not using Nourish at all (many druids seem to rely only on hots and Swiftmend, scarily enough).
Third, chat. Communication. This is and will always be the primary way of solving any kinds of problems. Talk with the group, acknowledge their frustrations and problems, and work with them to solve the problem you're facing.