It won't die by itself. Boner and the Republicans have to kill it, and they can do that whenever they wish since the House controls the purse strings.
So long as there is no Congressionally-approved budget, expenditures are controlled by the current services baseline budget and an assigned escalator.
That's exactly why Dingy Harry and the Senate have not submitted a budget for four years.
Under the circumstances, the ONLY control the House can have on expenditures is a.) the continuing resolution and b.) the debt limit -- neither of which allow for cutting any specific expenditures unless the Senate agrees. The House simply doesn't have the Constitutionally-assigned initiative regarding the purse strings.
Obviously, the Senate won't agree to any cuts of Obamacare.