So I was wondering. If I have an Omnath, Locus of Mana in play, no cards in hand, and no cards in play with activated abilites which cost mana. Is there a way for me to get rid of the green mana in order to shrink Omnath?

As far as I know, no. You'd have to be able to spend it somehow. I mean, if we're talking Magic: The Puzzzling you could have a Panoptic Mirror imprinted with Aura Finesse moving a Darksteel Mutation onto omnath and then end phase at which point your empties...

EDIT: Or there could be a TRIGGERED ability that lets you pay mana like an opponent's Rhystic Study or Propaganda

There are many potential mana outlets that aren't explicitly ruled out by your scenario. That is, there are many ways to spend mana that don't involve cards in your hand or activated abilities of permanents.

If you really meant "Can you just choose to remove mana from your mana pool without spending it on anything?", the answer is no.

Yeah, that's basically what i meant. Can I just dump the mana? I figured the answer was no but didn't know if there was some fringe ruling. Thanks!

In general, you can never do anything in a game of Magic "just because you want to". There has to be a rule, or a card, or an ability specifically allowing you to do that thing.

To paraphrase Richard Feynman, anything not expressly permitted is forbidden.