Does leveling up do damage?

Solution 1:

It most certainly does. I leveled up in the middle of a fairly large battle and it cleared the whole room of monsters in a huge explosion.

From the diablowiki page on Experience:

The actual level up moment is displayed very vividly in Diablo III. Instead of just a sound effect, there's a bright visual graphic, as a funnel of orange light washes down over the player, exploding outwards. Any monsters in range are blasted by the level up bonus, which deals considerable damage. In addition to that, there is a (very) prominent splash graphic that appears over the screen informing the player that they have leveled up, and will also fade in and out with other prompts telling them which skills they have unlocked, if any at all.

Solution 2:

Yes it does. And it also causes knockback, destroys objects and tosses corpses and clutter around a bit.