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:
They're also useful in Expeditions. For every "shiny" you pick up (tear or otherwise) you increase your chances of getting more Guild Quests when you finish.