900lb Gorilla I can appreciate, but "I've got a monkey on my back" - two opposable thumbs down...
Based on a first glance, it's looking like the original expression is "have a monkey on the roof," meaning a mortgage, dating back to the 1800s. Later "on the back" forms referred to any unpleasant burden, and in the 1940s began to be applied to narcotics addiction. No citations worth posting here, so the above is speculative.
I think the phrase predates references to a mortgage as a "monkey on the roof."
The earliest references to piggybacking monkeys involve the backs of other animals. Of these, the earliest I found were references to the cruel and ancient history of dancing bears, accounts of monkeys on baited ponies, and on show camels. In 1807, sportswriter Pierce Egan wrote of a monkey riding a horse. These seem relevant because of their proximity in time to the phrase first being used figuratively and because, in most of these anecdotes, the monkey is either frightening the animal or symbolizing its subjugation.
In 1822, Thomas Gillett published a collection of stories as The Midland Minstrel that included a humorous tale called "The Devil and the Doctor" about some students who teach their overzealous instructor a lesson by dropping a monkey on his back while he's riding his horse home one night. He's convinced the monkey is the devil and, after a mad dash through town, sees the error of his ways. This attribution of evil to the monkey is significant and evolves in later citations.
The earliest figurative use of the phrase I could find dates from an 1836 publication of The Parterre of Poetry and Historical Romance:
The italics were in the original, indicating, I think, the phrase was not in common use at the time. Its meaning is not entirely clear from its context of a man avoiding a pickpocket scheme at a fair. There may be a monetary debt connotation here. We do see a continuation of the monkey-as-devil theme in the use of diabolical.
Next I found what I think is the earliest printed definition of the phrase from the Glossary of Northamptonshire Words and Phrases, 1854:
And this from The Slang Dictionary, 1874:
Other literary references from the turn of the century reinforce this idea of the monkey as an evil spirit making someone do something they wouldn't otherwise, e.g, this from A Rational Marriage, 1899:
and this from Beauty and Nick, 1914:
The earliest mention of drug use I found was in a glossary from the 1938 Journal of Criminal Law and Criminology:
I believe that this slang for opiate addiction was a natural evolution of the phrase from its history of referring to an evil and subjugating force acting on someone beyond their control.
The original reference was to addiction, specifically to heroin, and there are several songs that use it in this way. These days people use it to refer to any compulsion or obligation, including good ones.