First off, it's pretty damn hard at the current gear level, even if you've got twenty-five-man raid gear. Some people love the challenge this presents, others hate the risk of frequent wipes that comes with it, others might feel a bit of both. Tanks are probably hit the hardest by this as they have the most difficult job in this instance (though everyone has to be on the ball), so don't be too hard on them if they mess up in some way. Instead be grateful for having a tank who doesn't instantly drop from the group as soon as someone dies, regardless of whose fault it was.
The instance basically consists of two parts, the Frostmourne event which results in you fighting Falric and Marwyn (technically two bosses, but since they are linked you might as well count them as one) and the escape from the Lich King. There's a single trash mob between them, but I doubt that should pose a serious problem to anyone.
The trickiest part of the first half are without a doubt the eight waves of trash mobs that you have to deal with. Since they spawn in random places all over the room and can't be targetted beforehand, it's quite tricky for the tank to pick them all up before they go and kill anyone. Most tanks like everyone to hide in one of the alcoves off to the side so all the mobs will run towards the same spot as soon as they spawn and the tank can intercept them, but if your tank prefers a different setup, don't reject it right away. After all he is the one who has to know what works best for him.
Now, as a healer you'll be staring at health bars most of the time and likely not be paying too much attention to who kills what, but I found that it really helps to know what's going on around you here (and I found out the hard way that not knowing what's happening can actually cause a wipe on its own).
Basically there are five types of trash mobs that spawn, a Ghostly Priest, a Phantom Mage (who can summon a mirror image of himself called Phantom Hallucination), a Tortured Rifleman (basically a hunter), a Shadowy Mercenary (rogue) and a Spectral Footman (warrior). I'm not sure exactly how many of each you get during each wave, but overall the group size and composition become more difficult to handle the closer you get to the end - multiple rogues, ouch!
The priest should be a high priority kill target, mostly because of the single target fear he has. If the healer ends up being feared just as the tank suffers from a huge damage spike, that can easily lead to a wipe. He also has an annoying AoE that does shadow damage and knocks you back, which can also be disruptive of any non-instant heals. In addition he can heal other mobs and casts shadow word: pain on your party.
Your dpsers' next target should probably the rogue, as he does typically annoying rogue things, like shadowstepping behind clothies and killing them before you can blink. He'll also poison and stun the tank, which will make said tank's health plummet like a rock when he suddenly loses all his avoidance. Be ready to spam hard and fast if you see this happening! In addition he throws poisoned daggers at random people, which leave a dot that ticks for quite a lot. If you can remove poisons you should do so whenever you get a chance.
The mage is annoying for the tank because if he doesn't have a ranged silence at hand, the only way to get the mage into melee range is the use of line of sight. Depending on your tank's class that can be a bit annoying for them and in the meantime the mob might randomly shoot other people. Fear not! The mage's fireballs and frostbolts don't actually hit for that much and it's not a problem if someone else "tanks" her for a bit. About the most dangerous thing she'll do is cast flamestrike occasionally, which will cause a fair amount of party damage, especially if people proceed to stand in it for its full duration. Also be aware of dpsers running out of your line of sight to whack the mage while she's standing outside of the alcove for example. You've got to keep your helpful "off-tanks" alive too!
The hunter is probably the most annoying mob to tank since he can't be silenced and is thus even harder to get into melee range. Some tanks will tell their dps to kill him early on as a result, but from a healer's point of view I very much have to disagree. The hunter's damage is pretty negligible and he can easily be off-tanked by someone else for a while. About the most annoying thing he does is use an ice shot on people which does a bit more damage and freezes them briefly. You'd think that this would be on the same scale of annoyingness as the priest's fear, but it only lasts two seconds, which is half the length of the priest's fear. His other abilities include putting down a slowing trap and cursing people so they take fifty percent more magic damage - which is why it's actually preferable not to have both the hunter and the mage on the tank at the same time! Like with the mage, you have to watch for line of sight issues when people decide to run towards the mob to get into melee range.
Lastly we have the footmen, who don't have much in the way of janky abilities, but simply hit very hard. It's very important that the tank picks these up without fail, as they'll pretty much gank anyone else within the blink of an eye, and even the best healer won't be able to do much about it. The one thing you really have to watch out for as a healer is their shield bash, which can be directed at any random target in melee range. I've had more than one wipe where the whole party stacked on the tank for easier mob pickup and then stayed there... until the healer got shield-bashed (and thus silenced) and then had to watch helplessly as the whole group wiped without heals. So it's ok to stand on top of your tank at the start to ensure that the mobs will come the right way, but as soon as he's grabbed aggro, make sure to take a few steps off to the side to ensure uninterrupted heals.
As soon as a new wave spawns, try to help with ensuring your own survival as well as you can. On my priest I found that fading as soon as I saw "entering combat" pop up on my screen did wonders to help direct the mobs away from my squishy behind. Paladins don't produce massive healing aggro and can survive a hit or two with their plate, but if something comes your way after all, don't be afraid of defending yourself. The bubble should only be a last resort; a quick hammer of justice or holy wrath should be enough to buy the tank a couple of valuable seconds to actually hit the mob that wanted to chew on you. Druids can place a HoT on themselves preemptively (while you're at full health it won't generate healing aggro anyway) and hit barkskin if they get attacked.
Mana can be an issue considering that there's a lot of damage going around and no time to drink until the boss is dead. Druids can always innervate themselves when they've used up about half of their mana, but for other classes good mana regen can be a matter of correct timing. As a holy paladin I found that the best time to hit divine plea seems to be when there are only one or two mobs left of the current wave, as incoming damage is relatively low then and the reduction to healing done doesn't hurt as much. If you actually wait until the last mob dies, divine plea might still be ticking by the time the next wave comes in, and you really don't want your healing done to be lowered then. For a priest that's also the right time to use hymn of hope if it's needed, while the shadowfiend should come out while there are still enough mobs around to give it something to hit for its full duration.
Finally, I noticed that I always run into trouble around wave eight or nine, whenever the double rogues show up. People start taking damage left and right and how am I supposed to keep everyone alive argh! This is the time to blow cooldowns. I'm sure I'm not the only one who sucks at using healing cooldowns because there's always this fear of "wasting" them and not having them available when I need them later. However, wiping on the eight or ninth wave in heroic HoR is an excellent teacher in that regard, because it's much more enjoyable to blow your cooldowns than to "save" them, wipe and have to do it all over again. If you're a priest,
The bosses that follow the trash are comparatively easy, but still not entirely facerollable. The most important thing about Falric is that he'll fear the whole party every so often (and since it's a horror effect there isn't much you can do about it) while doing a considerable amount of damage, and as his health goes down, everyone's damage and healing gets reduced by up to seventy-five percent, so be aware that you'll have to spam like crazy to get any significant healing in towards the end.
Marwyn is the more dangerous one of the two in my opinion, mainly because of his corrupted flesh ability, which reduces the target's health by fifty percent for eight seconds. Watch out for him casting this on the tank - I'll never forget that heroic HoR where I thought that we had it all in the bag, then he sliced the tank's health in half and hit him for twenty k damage - whoops, and it's a wipe. So make sure to keep your meat shield topped off at all times!
If you got any more helpful healing hints for this encounter, feel free to share.