Some of the new legendary gems, such as Mirinae, Teaerdrop of the Starweaver, cause elemental damage to occur (2000% weapon damage in this case). If I have +Holy damage on my gear, will this increase the amount of damage of this gem?
Yes, the elemental damage dealt by the gem ends up being increased by elemental damage gear.
The +Holy damage gear you have will not affect the gem's listed damage, but it will affect the end result. The gem will still hit for 2000% (or more depending on the rank). Leveling gems increases the value of the blue text on the gem, so this one will increase every level - not sure of the exact increase.
However, once the 2000% weapon damage as holy is calculated, the additional holy damage from your gear will be added on. Essentially, if you do 500,000 in white damage per hit, then this gem should hit for 10,000,000 in holy. If you have +20% holy damage, then the resulting hit should be 12,000,000.
Edit, some screen shots
Here is a normal white hit, as seen it is 290k
Here is a normal crit, 1.2m
The range seems to be roughly 250-350k on my character for white hits, and 800k-1.6m for crits. I am bringing this up because with 75% crit, the holy gem seems to only crit.
So, with 0% holy damage, this is what the crit from the gem looks like, 19m
And this is what the hit from the gem looks like (the crit was from my plain attack, the white is the gem), 3.9m
Okay so all of that was prior to the holy damage, I am now going to put on 20% and repeat the process:
As you can see, yes, the damage modifier does factor in, holy damage on gear increases the damage done by this gem.