I have heard the phrase "feeding the dragon" used to describe pouring time, resources, and energy into a situation that is self-perpetuating, caught in a positive feedback loop with negative consequences, or is growing out of control because of actions being taken. I think "giving a mouse a cookie" has similar meaning but with much less harsh connotations. Searches for the phrase online reveals many hits on Chinese economics and references to another phrase, "chasing the dragon" none of which shed some light on the meaning of "feeding the dragon". I have not been able to find much else. What is the etymology of this phrase?
I would interpret the phrase as a derivation of to feed the beast. The earliest Google Books results for this phrase refer to the literal feeding of animals, but by 1900 we have uses such as
in reference to keeping a potentially beastly person happy, or of subduing the beast within, representing the savage instincts of humanity that ration and civilization keep in check. That phrase may have originated in translations of Plato's Republic:
To feed the beast, then, is to surrender to something wild and uncontrollable. Perhaps you feed it just enough to stay quiet, hoping to tame it— but perhaps you indulge it, to make it stronger and more vicious (not unlike adding fuel to the fire).
You see results relating to China because the dragon is a synecdoche for China, which is being viewed as a metaphorical beast in some way (after all, feeding the panda or feeding the crane is not nearly as threatening). Similarly, you see uses like feed the bear for indulging Russia. To starve the beast is to end one's obeisance to the beast, enduring the consequences, in hopes of being freed of it; for fiscal conservatives, for example, to starve the beast is to deprive the government of revenue under the theory that it would force the government to reduce spending.
Chasing the dragon similarly uses a dragon to represent China, but is an unrelated phrase. It may refer to competing with China, or it may mean an impossible pursuit, or something else; it is hard to say without context.
The phrase appears to be a straightforward metaphor, as Dispensador observes. One early instance where the dragon really is a dragon occurs in Karl Stieler, Hans Wachenhusen, and Friedrich Hacklander, The Rhine From Its Source to the Sea (1878):
But the phrase "feed the dragon" has been used metaphorically even earlier. From "The Substitute" in The St. James Magazine and United Empire Review (December 1863 to March 1864):
The dragon here is a destructive engine that requires constant refreshment of human (and monetary) fuel, but is so fearsome that people take the path of least resistance in dealing with it. A one-word term for this policy might be appeasement, though there is some irony in applying that term (in this instance) to feeding a dragon of war, rather than to feeding a dragon against whom we fear to go to war.
Similarly, from "The Magical Cure," in The Health Reformer (May 1871):
In this case, the dragon is internal: an unhealthful combination of diet and inactivity.
In regards to Chinese economics, the meanings behind feeding the dragon and chasing the dragon refer to China as a dragon, its mythological mascot of sorts. When referring to China as a dragon, I feel it appropriate to capitalize the D. Feeding the Dragon would be aiding China's economy by giving it business and chasing the Dragon would be following in its footsteps, aspiring to the greatness that the country has achieved. You could take this further and say poke the Dragon for anything that would possibly agitate politicians or the military of the country, like if the US were to do nuclear weapons testing in the East China Sea for some inconceivable reason.
To actually go back to your initial question, feeding the dragon in the context of pouring time, energy, and money into something that keeps needing it, I would say that the etymology of the idiom comes purely from the connotations dragon has. As far as I am aware, within the cultures where English is spoken natively and dragons are part of the folklore, they are always fierce, brutish creatures. Feeding one of these dragons would only make it grow to be more fearsome. Someone with a drug addiction going off to do more of that very same drug would surely be feeding the dragon in that sense, cementing the extent of their addiction and making it harder to let go.
Maybe I'm thinking too bluntly, but there probably isn't more to the saying than that.
The etymology seems to be related to morality:
Dragonism is defined as "unremitting watchfulness" and there is a historical reference in clergyman Joshua Lacy Wilson's Episcopal Methodism; or Dragonism Exhibited (1811)
Wilson, a Presbyterian clergyman, and professor of "moral philosophy and logic" campaigned against theaters, dancing and Masonic order in the above-mentioned 'Dragonism' pamphlet. -"A companion biographical reference work to Who's who in America." Physical Description: v. 27 cm. (1607-1896)
The etymology may be in the oroborus (uroboros, oureboros) or the symbol depicting a dragon eating its own tail. - referred to in the Pistis Sophia (Gnostic text) (3rd -4th centuries AD)
I always thought that "feeding the dragon" came from the opium dens of Victorian England, where addicts would pour more and more time and money into their habit only to reinforce their addiction.
As mentioned in several other answers, "the dragon" part of the phrase came from the Chinese connection, although whether as a direct reference or because the dragon symbol was frequently used to suggest a "Chinese" parlor of this nature I'm not sure.