This has been tried and while it sounds good on paper, it probably would not work in this era of time. The powers of the Vice President are limited and it is pretty much left up to the President to determine what the Vice President will be doing while in office.
As the comment from http://wiki.answers.com which I have pasted below, it has happened several times. I remember from my college history classes, when being President did not involve a lot of international or major decisions that the Electoral College elected the President and VP from different parties. As the explanation below notes it would be much more difficult to accomplish under the present law.
Furthermore, with the ever growing gap between the two parties, I do not think it would work. I agree something needs to be done. I do not have the ultimate solution. I do know that it is probably more important that the two parties in Congress start spending more days in Washington looking for common ground and learning how to play in the same sandbox if you will. Party leaders used to be friendly toward each other and did work together. That is not the case today.
The Wiki Com comment is pasted below.
It has happened many times that the P and VP have been of different parties, for example, Thomas Jefferson and Aaron Burr, but, the possibility of this occurring has narrowed since the Twelfth Amendment stating that the P and VP should run for election on a "Joint Ticket". In theory it could still happen today due to the fact that in the Electoral College each "elector" votes for both a P and a VP, Faithless electors could simply vote for a P of one party and a VP of another President (however, many would be punished by state law). However, they could also abstain from voting for the VP which would allow the other Vice Presidential nominee to have more votes. An example of a Rouge or Faithless elector is in 2000, when Barbara Simmons refused to vote for Al Gore and Joe Lieberman as she pledged to do so, she did this so people would take notice of Washington DC's lack of representation.