Nekro's got it right, the best opponent against a bottom lane Morde is someone with a great sustained damage game.
Mordekaiser casts from health unless he has his shield up. Casting grants him his shield though, so once he starts going he's basically casting for free. In addition to this, having spellvamp means that his spells heal him in addition to providing him with a shield. Oh, and the whole point of this is to damage you.
So in the end, one spell does all of the following:
Damage your minions
Give him/increase his shield
has a net cost of 0
This is why you can't hope to outsutain a Morde. Plus, you can't outpush him because, as Nekro said, more minions just gives Morde more Metal Power. Your only option is to bully him nonstop and put much less of a focus on minions.
Talon, Graves, Cass, Urgot, and Kog'maw are all the sort of champs who can do this, and many more that I'm not thinking of off the top of my head. And fortunately, it works really well as compared to getting steamrolled with other strategies.
Just try to make sure your team understand that if the enemy has a bottom lane Morde, you need a champ who can keep up the damage pressure on him, or you can kiss your bottom lane goodbye.