Nightbane

[Lissa]

I'd agree with Frag in that I do not think that there is any case where there is an ability exclusive to one class. For example, Moroes is completely doable with zero priests: just bring skilled Paladins who know how to properly employ Turn Undead. For that matter, Moroes is tricky but even doable without any Priests or Paladins at all (but then he is difficult). It's easy to underestimate the range of capabilities available to a raid group.

For Aran, specifically, lacking a warlock is a significant loss but not immediately fatal. I don't know much about your group's composition, but it obviously has a hunter since you're in it. Scour the AH for some mail greens "of Frost Protection", then at ~42%, feign and macro-equip the entire set. Once the elementals pop, immediately shift focus to firing on the elementals and drawing their aggro towards you (click off Salv if you have it). Especially with a control strategy, there won't be any pressing need to have DPS on Aran while the elementals are out because you've presumably forced the poly earlier. With ~250 or greater frost resist (a paladin or shaman in your group can raise your resists easily, and a Magic Resistance potion works well if you don't have either of those), the elementals will all fire waterbolts at you and inflict very minimal damage. With all the stamina on the "Frost Protection" gear, you'll be very easily healable, and all you need to do is outaggro healing threat split five ways.

For groups lacking a hunter, a rogue can do the same trick with vanish, but being melee they're obviously disadvantaged. A paladin healing with Righteous Fury and employing Righteous Defense in moderate frost resist gear can also reliably draw aggro on all four elementals if no one else is touching them. For groups lacking hunters, rogues, and paladins, get a warrior or bear druid to gear up in frost resist from the start of the fight and have him run around sundering/clawing elementals once they spawn. If your warrior has trouble with it, get another one to do it too. You'd be employing a control strat again, which isn't DPS reliant, and you can interrupt in frost gear as well as you can in normal gear.

Now, having two warlocks makes it cake, and one warlock still helps a lot. Other solutions are obviously inferior, and I don't deny that. However, though other solutions are more difficult, that doesn't mean that they're impossible. It's not possible for Blizzard to tune an encounter to be equivalent difficult for all reasonable permutations for a 10 man raid group, but it is possible for there to be enough ways out of a jam that no raid group is completely screwed. One of those ways out is consumables, and on Nightbane it may be the only way out, which is distasteful in a number of ways. But every other fight has options.

I wouldn't say the issue is whether the fight is possible or not, but whether the difficulty is reduced. As has been noted, two warlocks on Aran makes it easy, three it's a joke. Two warlocks on Netherspite makes dealing with the blue beam easy, one makes it a little bit problematic when you have a non-warlock taking the beam, but not too bad, no warlocks means the healers have to be paying attention to who's in the blue beam. Nightbane is extremely difficult if you do not have a Paladin or Shaman along (easier for the Paladin because of more armor for damage reduction than the Shaman, but the Shaman can still get close to 50% damage reduction). Blizzard definitely designed Karazhan around having one of every class.

Ideally, I see the optimal Karazhan group as:

Warrior that has atleast 15 to 20 points in protection
Druid with a lot of feral talents and a good number of resto talents
Warlock any flavor
Mage any flavor
Priest that is either Holy/Discipline or Displine/Holy
Paladin that has a mix of Prot and Holy talents
Shaman that has a mix of Enchanement and Resto talents
Rogue any flavor
Hunter any flavor
and the filler being a Priest that is Shadow/Discipline.

edit: changed second Netherspite to Nightbane.