The Republicans want to play a dangerous game of chicken with the economy - I expect that there are a few at the margins who argue otherwise, but the consensus of economists on the issue is that a default triggered by the failure to raise the debt ceiling would result at best in severe harm to the economy and at worst in a return to worldwide recession. Republican demands are in part likely the same old game - get promises of future cuts that will never come to fruition in order to pound their chests at Tea Party rallies, even if they will later vote for budgets that don't include their own cuts. But, as Mitch McConnell has expressed, the larger goal appears to be to make President Obama a one-term President even if it means harming the economy and worsening the lot of working people.
Arguably President Obama has a greater duty to uphold the Fourteenth Amendment's mandate than he does to respect the debt ceiling - a limit that is being surpassed because the same Republicans who don't want to raise the limit previously approved spending in excess of that limit. If he does ignore it, I suppose some Republicans could attempt to sue him to force the treasury to default. But if I were the President I would be telling the Republicans who are endangering our nation's economic recovery and standing that I would be upholding the Constitution, as I am sworn to do, even if that means ignoring the debt ceiling. Then see who swerves off of the road.
Update: I think this article does a decent job of laying out the constitutional hurdles to ignoring the debt ceiling. Quoting Steve Bradbury:
I don't believe that the Executive Branch would be empowered by this provision in the Fourteenth Amendment to issue new debt in order to meet current interest payment obligations on previously issued debt, where the issuance of the new debt would cause the United States to exceed a statutory debt ceiling set by Congress, and I also don't think it would empower the Executive Branch to expend funds out of the Treasury on interest payments not covered by current appropriations of Congress.I think the argument here is that the President is not unilaterally issuing new debt, but is executing a budget passed by Congress that compels borrowing above the existing debt ceiling. If it comes down to ignoring the debt ceiling I suspect the issue is non-justiciable, so if Congress doesn't like the President's saving the nation's economy from its irresponsible, childish actions, its remedy would appear to be impeachment.