The administration has instituted regulations to undermine the law, for example expanding the sale of “short term” health insurance policies that do not have to cover pre-existing medical conditions or other health benefits deemed “essential” by the law, such as emergency services and prescription drugs. And the White House has cut funding to promote Obamacare sign-ups and for “navigators” who are supposed to help consumers enroll.

In seeking to get rid of the tax, the Democrats appear in one sense to be joining the slow dismantlement — or at least undermining their own argument that the embattled health law will save taxpayers more than it costs them over the long run while holding down health spending broadly.

But that proved to be of less concern than complaints from some of the Democrats’ key supporters, who say the Cadillac tax will hurt middle-class workers. The bill’s sponsor, Representative Joe Courtney of Connecticut, titled it the “Middle Class Health Benefits Tax Repeal Act” and recruited 367 sponsors, including 200 Democrats.

Even some liberal economists, backed by the unions, are urging repeal. The Economic Policy Institute, a liberal research group, argued that health care costs were already decelerating, so the tax was unnecessary both to pay for the health law and to further slow down health cost increases.

“To put it simply, the tax aims to reduce patients’ utilization of health care,” wrote Thea M. Lee, the president of the institute and a former leader of the A.F.L.-C.I.O., and the economist Josh Bivens. “But the glaring problem of U.S. health costs is not excess utilization; instead it is high and rising prices for health care. Smart cost-containment policy should address these prices, not seek to ratchet down how much care patients seek.”

With so many supporters, Mr. Courtney secured a vote on the House’s fast-track calendar, which requires two-thirds of the House to vote for passage, but also allows the bill to bypass rules that would require the cost to be offset by spending cuts or tax increases elsewhere.

Repealing the 40 percent excise tax on generous employer-sponsored health plans would increase projected federal deficits by $197 billion through 2029, according to the budget office, and so far there is no plan for replacing the lost revenue. Many health economists, who have generally been the biggest supporters of the tax, still see it as an important way to contain rising health care costs, contending that generous health benefits encourage people to get more medical care than they need.