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“I think Jim DeMint has had a very negative effect on the Heritage Foundation’s reputation for scholarship,” said Bruce Bartlett, who served as an adviser to President Reagan.

Mr. DeMint last week concluded a nine-city “Defund Obamacare” tour organized by Heritage Action for America, his group’s sister organization, finishing with a packed rally in Wilmington, Del.

Mr. DeMint joked that Heritage organizers had organized the standing-room-only crowd to “simulate what it’s going to be like in a doctor’s office in about a year or so” when the president’s health care law fully takes effect.

Mr. DeMint is calling on grass-roots conservatives to oust Republican lawmakers who refuse to fully embrace the push to defund Obamacare — even if that leads to a government shutdown, which some Republicans warn could harm the party’s chances come Election Day.

“I think [President Obama] knows that Republicans are afraid, and if they are, they need to be replaced,” Mr. DeMint told National Public Radio last week.

The effort has garnered a lot of media attention, though just 14 of the Senate’s 46 Republicans have pledged to reject any spending bill that includes funding for the law.

Losing clout?

Some analysts wonder whether Mr. DeMint has been diminished by leaving the Capitol. He surprised his colleagues and the political world by resigning his seat in December, two years into his second six-year term in the Senate, to take the Heritage post.

“I don’t think he has the influence he had in the Senate,” said a top Republican official. “I don’t think he is the difference-maker because he is just not in the mix.”

His hardball tactics even might have led to friction with the strongly conservative House Republican caucus. The National Journal reported last week that Heritage officials were disinvited from attending private strategy sessions of the conservative House Republican Study Group after the think tank harshly criticized the farm bill passed by the House.

The defund Obamacare tour, which started in Arkansas in mid-August, put Mr. DeMint at odds with a number his former Republican colleagues in the Senate, including fellow South Carolinian Sen. Lindsey Graham.

Mr. Graham, who is up for re-election next year and could face his own conservative challenger in the Republican primary, has said that he wants to get rid of Obamacare but warns that shutting down the government would hurt Social Security recipients and military funding.

Mr. Toomey and Mr. Johnson also have been reluctant to sign off on the effort, while some insurgent candidates are backing the push, including Nancy Mace, one of Mr. Graham’s opponents in South Carolina, and Matt Bevin, who is challenging Senate Minority Leader Mitch McConnell in the Republican primary in Kentucky.

Heritage Action, meanwhile, announced last month a $550,000 ad campaign that will run in 100 districts represented by Republicans — including House Speaker John A. Boehner of Ohio and House Majority Leader Eric Cantor of Virginia — who have refused to “take the steps necessary to defund Obamacare in its entirety, including on a year-end funding bill like a continuation resolution.”

The Senate Conservatives Fund, which Mr. DeMint led until this year, is running radio ads targeting Republicans including Sens. Thad Cochran of Mississippi, Lamar Alexander of Tennessee and Richard Burr of North Carolina.