I heard from somewhere that gods maintain their power from the people worshiping them. The more people pray, the stronger they get. I also heard that If their home/environment is destroyed (eg. Posideon's waters being polluted) then they will lose some of their power.
What I would like to know is...
Is this true?
Can they be killed if no one cares about them?
While I don't know of any Greek or Roman sources that discuss this topic in complete seriousness, it does appear in the comedy of Aristophanes (fifth century BCE)—which indicates that the idea existed in the public consciousness to some extent.
In the Birds (aka Ornithes in Greek, Avēs in Latin), an army of birds and transformed humans bands together to overthrow the Olympians and take their place. Their plan relies on building a floating city called Nephelococcygia ("Cloudcuckooland"), from which they can blockade Olympus, preventing human offerings from reaching the gods.
This scheme does in fact work, and the gods promptly begin to starve for lack of sacrifices. It's implied that they're literally starving to death (the whole thing is a reference to the Athenian blockade of Mēlos a few years earlier), and in the end Zeus is willing to give up pretty much anything (including a lot of his godly power) in exchange for ending the blockade.
Now, Aristophanes is a comedian first and foremost, and everything he writes about mythology should be taken with many grains of salt. Nevertheless, this does show that Greek authors were thinking about this idea well before Christianity took over.
I don't know about how true it is -- really, how can we? But it is an ancient idea, first put forth by Plutarch, who posited that Pan had died because people no longer believed him to be more than a story (In "The Obsolescence of Oracles"):
The power of the spirit does not affect all persons nor the same persons always in the same way, but it only supplies an enkindling and an inception, as has been said for them that are in a proper state to be affected and to undergo the change. The power comes from the gods and demigods, but, for all that, it is not unfailing nor imperishable nor ageless, lasting into that infinite time by which all things between earth and moon become wearied out, according to our reasoning. And there are some who assert that the things above the moon also do not abide, but give out as they confront the everlasting and infinite, and undergo continual transmutations and rebirths."(section 51)
(That idea becomes a subplot in The Percy Jackson series).
Carl Jung said “The gods have become diseases; Zeus no longer rules Olympus but rather the solar plexus, and produces curious specimens for the doctor's consulting room, or disorders the brains of politicians and journalists who unwittingly let loose psychic epidemics on the world”
God’s can’t be killed as such and mortal humans can’t increase or decrease a god’s own personal, innate power with physical acts. When forgotten, they merely fall below the threshold of human consciousness and act on humanity via the unconscious as per Jung’s quote above. Your question presupposes a very literal and physical existence for the gods whereas mythology is really allegorical and their true nature is more psychological (or spiritual) - although no less real for being so.
Also, the story of Pan’s death may well have arisen from a mistranslation of the Adonis ritual where the celebrants re-enact his life and death and shout out at some point, Adonis is dead. This was erroneously translated as Great Pan is dead which seemed to chime with a general belief at the time that the pagan religions were in terminal decline.
