Occasionally when a monster drops a shiny present and you pick it up, it is a "tear". But I can't find it in my inventory nor item box. It also doesn't seem to be in any combination that I am aware of.
Are tears just "junk" items that the game immediately throws away but lets you feel special by picking one up?
Even the Kuranico doesn't really say what they are used for, and they aren't even listed in the Wikia.
Best Answer
Tears are account items, which means they'll be turned in at the end of the quest for money. They'll occasionally be a subquest as well, in which case you need to turn them into the supply box before the end of the quest. With a few exceptions (like Gypceros), the only way to get a tear is via a shiny drop. Every large monster will drop an item under certain conditions, but the conditions differ from monster to monster and the drops are usually limited. For example, in MH3U you could make Lucent Nargacuga drop a shiny by throwing a sonic bomb at it when it readies itself for a lunge attack, but it will only drop a shiny twice. Try it a third time, and nothing will drop.
The one exception is the item Basarios Tears, which is an item you get from Basarios and Ruby Basarios which is used in crafting.
Something I forgot about, which was pointed out in the comments by Pyritie: