While on its face your plan seems simple, you're omitting a few challenges.  
Poor people have fewer options to increase their efficiency.  They live in cheap, shabby apartments with abysmal energy efficiency, but that's all they can afford.  Wealthier people weatherize their homes and buy LED lightbulbs.
They drive 1993 pickup trucks with 130,000 miles that get 9 mpg, because that's all they can afford.  Wealthier people buy latest energy efficient cars.
As energy conservation goes up, tax revenues go down, and the government is forced to raise the tax higher.
Taxing BTUs is fine, but the revenue shouldn't go into the general fund.  It should be reinvested in helping people become more efficient.