Oh man, I'm gonna have to keep trotting this out, huh?
Short answer: they can't.
Long answer: removing tea partiers basically hastened their party's status to Whig status since
http://www.newrepublic.com/article/115189/gop-cant-survive-without-tea-party
Poison pill A or B: They'll be dead thanks to Tea Partiers in a decade or so, or dead now if they get rid almost half of their electorates.
To go back a little bit, the posts about why the GOP doesn't lose the Tea Party are all solid, but I wanted to quickly respond to this, because I think it misses a very important component of why America has a two-party system.
tl;dr The Republican Party isn't going anywhere -- it's just going to lose everything for thirty years and then reinvent itself.
In a first-past-the-post election system like America, there's enormous pressure towards the median voter and towards combination of parties. What isn't as clear is that these parties don't have to align along any specific axis or policy. An American political party actually represents a temporary coalition of interests that can unify on certain common goals and, when in power, provide each other support for their specific goals.
Generally, one coalition is dominant, meaning that they can consistently command half the voting population, and wins most of the time. The other coalition spends its time attempting to peel off members of the dominant coalition by running candidates that can appeal to specific interests in that coalition and that offer a policy platform that adheres in the broad details to that of the dominant coalition. In this way the dominant coalition controls the "political conversation."
Already in this description you can probably recognize the dominant political coalition from 1968 to around 2006 -- the Republican Southern strategy coalition, organized around libertarians, Randians, social reactionaries and bigots, and corporate interests, all of whom can unify on the need for reduced taxes and reduced federal power for their own purposes. This coalition replaced the old Republican coalition, which I'm not really sure what it was actually, and became ascendant over the previously dominant Democratic New Deal coalition.
When the New Deal coalition (broadly, organized labor, progressive Christians, people of color, rich white liberals and the poor) started losing its grip, you saw a lot of drama. Extreme elements in the Democratic party started taking center stage, because the establishment no longer had the power to control them. They started pushing for policies badly out of step with the American mainstream, predicting revolutionary changes, and engaging in unpopular and dangerous unconventional tactics. You've probably seen this on television, assuming you were watching Mad Men last season. But these were all fundamentally death throes of a coalition whose members knew it was coming to an end and wanted to get what they could out of it before the party was over.
After this the Democratic party spent decades in the wilderness. Major components of their former coalition were jettisoned -- labor unions will never again have the influence they did in the New Deal era, and organized progressive Christian movements essentially no longer exist. When they did win elections, they did so by running Southern candidates to appeal to the Southern cultural voters and by advocating a lessened version of the same "less taxes, less spending" argument that the dominant coalition favored. In this way the Southern strategy controlled the political conversation for almost fifty years.
But in 2008, as you might have noticed, things changed. The New Deal coalition was replaced by a new coalition -- people of color, women, GLBT, Millenials, and rich white liberals -- which has the uninspiring name of the Rising American Electorate. This is the new Democratic coalition, and it's the dominant coalition of American politics today, focused on social progressiveness, urban regrowth, and increased government social programs.
If the above story of party reinvention seemed awfully familiar to you, it might be because the GOP is going through the same thing right now. Extreme and dangerous tactics? Check. All-or-nothing policies that have no support? Check. Nominating candidates specifically to appeal to the dominant coalition? Well, not yet, but check back with Marco Rubio in 2016. Jettisoning components of its coalition? It's happening already.
So I don't think the Grand Old Party is going anywhere. It's not going to win much for the next 20-30 years, and when it does, it's probably going to be somebody more like Carter and less like Reagan. But it's going to learn what works and reinvent itself. My bet is that social reactionaries are going to lose.