Impeached and convicted or not, the constitutional crisis would be draconian in the extreme. One possible solution would be for President Obama to resign in order to spare the country lots of turmoil. Then, or if President Obama were removed by the impeachment process, Vice President Biden would become the president, making Speaker Pelosi next in line. This might well be even worse than having President Obama remain in office. Alternatively, President Obama might simply remain in office, a much weakened "lame duck" president, quite probably facing an opposition Congress following the 2010 elections. That might well be the best possible outcome.
There is at least one other possibility, although it holds at best modest facial appeal. All of the laws enacted since 20 January might be attacked as invalid, because signed by a spurious president. Aside from the fact that these attacks would take a long time to make their tortured ways through the courts, they would, if successful, cause a big mess. What about appropriations bills? The Government needs funding. We may joke that it would be neat if the Government simply stopped, but realistically that would have some pretty undesirable consequences: the military needs to pay troops and even to feed and arm some of them. Could troops remain active in Iraq and elsewhere? Could they even be brought home? Perhaps they could hitchhike. Like it or not, the Government needs to function, and this requires such mundane things as electricity, salaries, aircraft fuel, and lots more, all of which costs money.
No matter what happens, President Obama will most likely remain in office at least until 20 January 2013. There is a useful teaching point in all of this, however. The "fourth estate," the press, let the country down very badly during the primaries and general election campaigns. Rather than pursue even tepidly the question of Candidate Obama's constitutional qualifications, the press grossly disparaged anyone who raised such questions and made sick jokes of the whole thing. That spectacle continues. If a valid Kenyan birth certificate were to surface, it might wake up the press; if that were not to do so, then nothing would.
There are a few things which might be done to avoid similar crises in the future. The states could require proof of constitutional qualification before any candidacy is certified. There are constitutional qualifications for all federal elected officials, and the constitutions of many states likely also specify qualifications for office. There is a move afoot to push for a federal law so providing. The political parties could impose requirements that prior to primaries, candidates make reasonable showings, under oath, of their constitutional qualifications. The Congress might some day pass legislation criminalizing the assumption of the presidency knowing that one is not constitutionally qualified. Maybe some day, another president might sign such legislation, making it the law of the land. Until now, these have been pipe dreams. Now, in the probably unlikely event that a valid Kenyan birth surfaces, there may be sufficient impetus.