Does armor affect shield health?

So for this, we tested using Zarya's self-damage on her secondary fire since we only had two people. Let's establish the expected behavior:

No armor.

That's 23 damage dealt to Zarya without armor.

Now, let's get to the meat of the question.

With armor, damage on shields.

This is also 23 damage dealt to Zarya's shields while she has armor from Torbjorn's armor pack. But let's give it one more trial.

With armor, damage on armor.

269 - 254 = 15 damage dealt to her armor. While this is a bit more than the expected 5 damage reduction, it's still reduced damage which means that self-damage is mitigated by armor.

Conclusion

This means that we can definitively state that, at least for now (6/30/2016), the damage reduction provided by armor does not apply to shields.

According to the OverWatch wikia it is covered that there won't be a reduction even if there is Armour on your character: Shields don't reduce damage (even if you also have armor)

http://overwatch.gamepedia.com/Hit_points#Armor

Someone asked for science. According to this guy who did this already it was a misconception that people believed:

He covers this around 3:20 or covers more of it over the entire video.