A Palin/Cheney presidency would hasten a lot of very bad things, but assuming that it will cause the "end of the world" is a bit much. 

Their energy policy would hasten climate change and kill all possible reform of it, but even in the worst legitimate scenarios, only a fraction of the world would be ruined by flooding, heat, droughts, tornados, etc., and probably not on a large scale until the 2030's.  

Yes, we would hasten major food and water shortages, but this would again only kill a fraction of the world's population within the next 30 years.  It would hasten the extinction of certain species, etc, and decimate a lot more natural resources, but we wouldn't feel those effects for a long time.

Yes, we would probably see some major global fireworks as regarding Iran, North Korea, Pakistan and Russia, and we might see a nuclear conflict between Iran and Israel, which would kill millions of people and probably lead to a fairly drastic re-drawing of certain Middle Eastern borders, but projections of cataclysmic global nuclear annihilation seem far-fetched.

And we would almost surely see the demise of America as a superpower, being a) completely bankrupt, b) rejected in any global initiative by the rest of the world, c) an utter laughingstock to the other civilized countries in the world, and d) no longer a place that welcomed much civil discussion on any matter of policy.

But, to be honest, a lot of these things are true today already, or are (seemingly) inevitably on that path. Look at things today--the government gets nothing done, the president seems to have almost no power of persuasion with Congress, religious lunacy and lobbyists and war-mongerers and know-nothings already dominate our political landscape and wield most of the power over our country, and we are already broke, already procrastinating the major crises the globe soon faces, and already crumbling morally, structurally, educationally and in terms of our competitiveness in the international workforce.  America is a dying empire, and it might seem like the end of the world because of how mythically exceptionalist we are, but Europe and Asia and Latin America would carry on with their own existences even if there was a global financial depression.

Bush and Cheney apparently (at least I cannot find evidence to the contrary) tried their hardest to screw up every single agenda item (except foreign aid) as thoroughly as possible.  And, while we might not be living much longer in what our parents saw as the "shining city upon a hill" that used to be America, and while almost every global and national trend has gotten worse since 2001, the world didn't end.  And it won't end with another horrible president, even one who is more arrogant and unqualified as Bush was.  And I think we have probably seen the Dick Cheney show play itself out.  It isn't like 9/11 happened in 2008 and we barely caught a glimpse of what an unscrupulous authoritarian war profiteer could do as Vice President.  Hard to imagine he could do much more damage than aiding in us ripping up our Constitution, going into two ill-advised trillion-dollar endless wars, emboldening every enemy we have on the planet, appointing two Conservative robots to the Supreme Court, drowning an American city, crashing the entire financial system, selling us out to Big Pharma and Big Oil, decimating the science, education and arts budgets, and never finding Bin Laden.

So this would be another dark moment in the 30+ year period of great darkness that has enveloped the United States.  But Palin/Cheney would just be a 10% worse version of the same.  And December 21st, 2012 will just be a slightly more paranoid recurrence of the Y2K scare.