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
They perfectly understand the utility of “feeding the beast” with a nice dinner to keep him good-tempered
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:
I mean those which are awake when the reasoning and human and ruling power is asleep; then the wild beast within us, gorged with meat or drink, starts up and having shaken off sleep, goes forth to satisfy his desires; and there is no conceivable folly or crime … which at such a time, when he has parted company with all shame and sense, a man may not be ready to commit.
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):
Another version of the legend [of the Drachenfels] is, that by the advice of their priests, the pagans of the mountain were in the habit of feeding the dragon with the bodies of their prisoners, and in order to keep him in good humour they were obliged constantly to provide him with fresh victims.
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 levy was being vigorously pushed on, at that very time, throughout the north-eastern States, and Colonel Zerubbabel Wilks, with his regiment, had arrived in New London for the purpose of enforcing the draft, in case of resistance being offered to the Commissioners. Men must be had to feed the dragon of war, and bounties had long since failed of their primitive effect. There had been much growling and talk of armed opposition on the part of the labouring population of the little seaport, but the presence of Wilks and his Zouaves, coupled with the knowledge that Colonel Schurtzer and his marauding regiment of Germans, renowned for their lax discipline and rough treatment of civilians, were encamped on the banks of the Connecticut River, caused the discontent to evaporate in harmless grumbling.
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):
My good friend, you are in a terrible situation ; but I can help you if you will follow my directions. You have a horrible animal in your stomach—a dragon with seven mouths. I must talk with the dragon myself, face to face, so you must come to me. But in the first place you must on no account either drive in a carriage or ride on horseback—you must travel on the shoemaker's nags ; otherwise you will disturb the dragon and he will devour your intestines in his anger. In the second place, you dare not eat anything but the simplest food ; in the morning, a little soup with vegetables sliced in it ; at mid-day, a sausage and one plate of vegetables ; the same at evening, only an egg ion place of the sausage. Whatever else you may eat will only feed the dragon, who will grow larger, and your tailor will very soon be obliged to yield his place to the undertaker.
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.
IT seems to "feed the dragon" is a metaphor for supplying more fuel to that which does not serve us. And yet, a dragon is not a beast, nor is it a negative, it's a myth, a metaphor. Dragon's have power, if the dragon is within the psyche it needs food to thrive to be powerful and must be fed.
We fear the dragon, and we resist feeding it, we restrict the flow to the dragon and the dragon becomes the latent hunger that drives the behaviors, but this is not the dragon, its the result of not feeding, purposefully avoiding feeding the dragon, restricting the dragon at ones own peril.
During our drumming journey tonight, several people had a similar experience with a dragon. All of us actually, and the dragon left a clear message with one to "Feed your dragon".
The evidence to restricting life's flow would unbalance and drive compulsive behavior... as will any restriction. A happy dragon would be one that holds power and is regularly fed and with no fear of being denied what it most needs - it's food.
To starve a dragon would seem to unleash the monster, and at your peril on your own head be it.