I realize this question is old, but I feel compelled to add in here.
In addition to what's already been said (boost your MR, don't be afraid to over-nuke him to eat past his shield), there is the topic of range and escape abilities.
Mord is a unique melee in that his abilities have some good AOE and range to them, but he suffers the penalty of normal melee characters in that a good ranged will still be able to keep him at bay. Characters with long range and/or skill shot abilities like Anivia, Ezreal, Kog'maw or Karthus can put pressure on Mord very easily during the laning phase, which is where Mord tends to establish himself. A character with strong lane presence/harassing ability like Nasus, Garen, Vladimir or Fiddlesticks can also hurt Mord's early game by keeping him at the enemy tower where the tower will eat most of the kills.
Mord also suffers from a distinct lack of escape abilities. Ganking Mord can be easy with a properly built tank to bait his abilities into cooldown, and then bum rushing him when he has nothing left with which to counter. Slows and stuns prevent his otherwise relatively high HP and shield from making a difference, so plan ahead.
Mord tends to be very item dependent. A tank Mord ends up having lower overall damage, while a DPS Mord ends up having a very low lifepool and no escape mechanisms. Putting pressure on him in his lane early game prevents him from racking up too much money (which is why he was designed with so much AOE ability), and thus prevents him from getting his really good items at the proper time which would otherwise allow him to become the "powerhouse" that everyone fears.
Don't get me wrong, a properly played Mord is quite good, and a smart player knows how to balance between him being a tank and a burst DPS. But a heavily pressured Mord will crumble late game, and a ganked Mord will have no chance against the proper lineup.
Enemy: Heavy AD - Rammus, Malphite, Amumu
~ Rammus' passive naturally scales from armour, giving him higher auto attack damage
~ Malphite has the ability to slow auto attack speed of enemies
~ Amumu's ultimate renders those enemies affected unable to attack or move for two seconds. Additionally his ability, Tantrum, reduces physical damage received by a flat amount (calculated after the reduction from armor), and the ability's cooldown is reduced by 0.5 seconds for each time Amumu is hit by an attack.
Enemy: Heavy AP - Galio, Mundo
~ Galio naturally scales off of Magic Resistance
~ Mundo has amazing HP regeneration, something that is hard to counter for casters, especially seeing as if they have no anti healing spells (such as katarina), the only feasible counter is the summoner spell ignite.
Enemy: Heavy AoE - Maokai
~ Maokai has an damage reducing AoE
ult, a great way to counter enemy AoE
spells
Enemy: Tanky Team - Cho'Gath, Mundo
~
Cho'gath gets seriously beefy if his
ult has been used well. It can give
tanks a big surprise due to the only
tank having true damage.
~ Mundo with his health regeneration is going to be a very hard guy to take down if the enemy lakes high DPS. His Infected Cleaver ability also deals magic damage equal to a percent of the target's current hitpoints, making him an effective harasser and damage dealer against many tanks.
Enemy: Low mobility - Singed
~
Singed has extremely impressive mobility and can be an amazing initiator / disruptor with his move speed, AoE slow and fling.
Enemy: High mobility - Shen
~ Shen has a global ultimate that
can directly counter enemy global ultimates, such as TF's and Pantheon's, shielding the target they are attacking and teleporting to their side to assist them.
Enemy: High CC - Alistar, Mundo
~ Alistar's ult removes all CC when cast and can be cast through stuns, silences, and the like.
~ Mundo's Burning Agony ability reduces the duration of stuns and slows on Mundo by up to 35% at level 5.
Friendly: Heavy AoE - Amumu, Galio, Nunu
~ Amumu has an AoE root, which sets up
AoE ultimates on your team perfectly
~ Galio has an AoE taunt that damages. Nearly as good as Amumu's root, perfect for setting up AoE ultimates
~ Nunu has a -devestating- ultimate if used correctly. In combination with some AoE CC, his ultimate slows and burst huge damage if allowed to channel to the maximum.
Friendly: Lack CC - Alistair, Cho'Gath, Amumu
~ Alistair is a CC machine, his
stuns/knock backs are not effected by
any CC reduction
~ Cho has heavy CC, with his AoE knock up and silence.
~ Amumu has a stun with a relatively low cooldown (however it is a skillshot) and an AoE root, with a pretty large radius
Friendly: No jungler - Nunu, Amumu
~ If you need to jungle as a tank,
Nunu is probably the best roamer and
jungler combined, Amumu is the second
most capable jungling tank.
Best initiators - Amumu, Malphite, Rammus, Shen, Singed, Galio
~ Amumu - Bandage toss into the enemy, and initiate fight with ultimate
~ Malphite - AoE stun, with a good range, and an auto attack speed debuff
~ Rammus - Roll into enemy carry, stun, taunt them into your team
~ Shen - Shield, AoE taunt.
~ Singed - Flip a squishy into your team and AoE slow the enemy
~ Galio - Flash and ultimate into the enemy
Oh course the tanks I have mentioned can fill the role of tank vs any type of team, only they are specialised in the areas I have mentioned.
Best Answer
If they have a lot of HP, have your ranged champion(s) shred away at it with Kitae's Bloodrazor ( deals 2.5% of the target's maximum HP as magic damage -- basically, it's a weaker version of Madred's Bloodrazor. more info here: http://www.mobafire.com/league-of-legends/item/kitaes-bloodrazor-128 ). This should be purchased by a ranged champion with a high attack speed. Hit-and-run harass them until they're too low to defend (or until you can score a kill on them).
Kog'Maw would also be an effective counter, especially with Kitae's Bloodrazor, since his W skill is essentially a Bloodrazor in itself, and because his longer range (and great synergy with items like Frozen Mallet) will allow him to harass even harder. And it goes without saying, but his ultimate is another great way to chip away at / drive out those champions who like to camp at their turrets (also note that his ultimate is great for defending as well, since a single hit from far away will stop a capture).