How do I get rid of Zenyatta's Orb of Discord?

As the other answers have said, killing the Zenyatta removes the orb, as does Recalling with Tracer or going invulnerable as Reaper (Wraith Form) or Mei (Cryo-Freeze).

Additionally, Zarya's Projected Barrier and the personal Barrier both remove the Discord Orb, as far as I know. You can call out to an allied Zarya or play her yourself to remove it.

This one is the most situational, but using Self-Destruct on D. Va, getting your mech killed, or calling down a mech, will remove the discord orb from you.

LOS'ing the Zenyatta is the most effective and reliable strategy, however.

According to the wiki,

The orb of discord returns after 3 seconds if not in vision.

So to get rid of it, you can run out of his line of sight for 3 seconds. Alternatively dying also removes the orb of discord, but is somewhat counter-productive. Also, if you kill the Zenyatta, the orb will go away, though it can be risky since you will take extra damage while fighting him.

Note: An orb of discord cannot be placed on someone behind a shield, so if you stay behind one, you won't have to worry about getting one.

Also, as @uni and @yuuki mentioned, some abilities can remove the orb from you.

Tracer's Recall ability removes Zenyatta's Orb of Discord and, technically, Mei's Ice Wall can remove it by breaking line-of-sight. Reaper's Wraith Form will also remove Orb of Discord.