While skilling and killing monsters, some mysterious divine tears/anagogic orts appear in my inventory or as part of the monster loot. What are they? And what do they do?
If you are part of a clan with an Clan Avatar, you will have a chance of obtaining anagogic orts from skilling or from monsters. With 300 orts, you can interact with your Clan Avatar to receive one of seven special buffs for a week (Credit to the (unofficial) RuneScape Wiki for the following info):
Note that you can only receive 200 orts a day.
Also, many players consider getting the ort drop as annoying. There is a way to disable getting the orts as a drop. Talk to the Captain of the Guard at the Clan Camp (Easily reachable with a Clan Vexillum).
Divine tears act as the same way as orts (except they appear in your inventory instead of on the ground, unless they are found from combat). However, their use is completely different.
With RuneScape 3, the Battle of Lumbridge now occurs. If you follow either Saradomin or Zamorak, you can collect divine tears from the battleground (and also from skilling/monster drops). You can deposit these at collection points beside either God. You deposit these to help your God defeat the other God, obtaining the energy underneath them.
As a reward for depositing the tears, you get to vote on a buff which will be activated in approximately 2 days on 17 hours since posting this answer (the amount of votes you get is 1/10th of the amount of tears you deposit). Also, you will obtain renown (the amount is the same as the amount of tears deposited), which can be used to purchase level 75 hybrid armour, experience lamps, or even emotes and titles.
You can only obtain 3,000 tears a day, or 1,320 tears if you are free-to-play.