Substantively? Not much. But he could possibly look damned good doing it. That could be a problem for the GOP.
Shooting straight from the hip I would think that impeaching Obama would piss off more people than the GOP can afford to piss off. Don't mistake the GOP win for a big "like" from America, it wasn't. The GOP doesn't have that much political capital to gamble on an impeachment hearing. Then there is the minority vote, which the party of angry old white dudes needs and will need more and more of in the future. I'm not an ethnic minority and thus I cannot speak for or with any real insight into ethnic communities, but I don't believe they are in love with either major party. I know for certain they aren't head over heels in love with the GOP. Many may equate impeachment with lack of respect, commitment, understanding of ethnic issues. Does the GOP really want to take that chance? Hell no, they don't.
Have it all happen and problems over a Ferguson verdict, minority voters just aren't going to flock to the GOP. Then Biden becomes the President and doesn't screw anything up and looks good and trustworthy. It could happen. Biden isn't stupid. He's not diplomatic, LOL, but that could play in his favor,not being slick and polished at a time when most of us are sick of that crap.
The establishment GOP is having none of that. They are better off having Obama finish his term and I think the establishment GOP are very much aware of that. Tea baggers? I don't know. Neocons? I have no idea. Krauthammer is in some ways like Limbaugh, except he is more educated.
Also, the Demos have their own civil war going. How many will view a blatant disregard for the Constitution and our way of government and not act?