Question: 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?

Answer: 
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 

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> They perfectly understand the utility of “[feeding the beast](http://books.google.com/books?id=oJhyAAAAMAAJ&pg=PA245#v=onepage&q&f=false)” with a nice dinner to keep him good-tempered
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in reference to keeping a potentially [beastly](http://www.collinsdictionry.com/dictionary/english/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](http://classics.mit.edu/Plato/republic.10.ix.html) translations of Plato's *Republic*: 

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> 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. 
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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](http://www.the-american-interest.com/articles/2007/03/01/dont-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](https://en.wikipedia.org/wiki/Starve_the_beast)* is to deprive the government of revenue under the theory that it would force the government to reduce spending.

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*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.
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.