The plan is to oppose any bill to fund the government or increase the debt limit that also provides money for putting the health-care law in place. Because Republicans control the House, Democrats can't continue borrowing or paying for government operations without Republican support. So, conservatives say, Republicans should insist on defunding Obamacare as the price of that support.
The chance that Democrats would go along — would give up on their signature legislative initiative of the last decade soon after having won the presidential election and gained Senate and House seats — approaches zero percent. So if Republicans stay firm in this demand, the result will be either a government shutdown or a partial shutdown combined with a debt default.
Either would be highly unpopular, and each party would blame the other. The public, however, would almost certainly blame Republicans.
First, Republicans are less popular than the Democrats and thus all else equal will lose partisan finger-pointing contests. Second, the executive has natural advantages over a group of legislators in a crisis atmosphere. Third, people will be naturally inclined to assume that the more anti-government party must be responsible. Fourth, some Republicans will say that government shutdowns or defaults are just what the country needs, and those quotes will affect the image of all Republicans. And fifth, the news media will surely side with the Democrats.