You don't need a healer.

Every class can solo through the game so any same-class duo can and will work well. You may not get the optimal synergy depending on the class you choose. I'd stay away from two melee char just cause positioning can be a pain. Two hunters and two warlocks have great synergy though. Two shadow priests would work great too. Mages would be good if you spec'd for mana efficiency. A feral/caster druid would be great.

If you want to mix up the classes, I'd worry about what classes you want to play, which you can control, and how well they synergise. The only tough pairing would be two-melee just because of control problems. I'd maybe avoid contrasting styles like mage/warlock because one is DoT and the other is DD based. Some classes are awesome synergy like druid/rogue because they cover each others weaknesses brilliantly with no downside (you get dual-stealth).