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: 
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.
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.