The US Constitution says the Vice President of the United States may only vote in the Senate to break a tie.
The Vice President of the United States shall be President of the Senate, but shall have no Vote, unless they be equally divided.
The Constitution of the United States, Article 1, Section 3
Vice Presidents have used that power to confirm appointments and vote in favor of bills that the President wants. The Vice President is the only officer of the U.S. government with both an executive and legislative role. As somebody who is elected along with the President, it would seem unthinkable for a Vice President to vote against his (maybe soon someday, her) own administration.
Has any Vice President voted against a bill or nomination favored by the President?
Some situations where this might have occurred was early in U.S. history when the Vice President and President were occasionally members of different parties or political rivals. Vice President Thomas Jefferson was a rival of and member of a different party than President Adams. The other occurrence was when John C. Calhoun was Vice President and John Quincy Adams was President. Both were members of the Democratic-Republican Party, but that was the sole major political party at the time, and each man was part of a different faction within that party.
Jefferson voted 3 times to break ties in the Senate and Calhoun voted 31 times. Did either of them vote against the administration? (At that time, the Vice President was not as integrated into the administration as it is now.)
There's also the possibility, albeit unlikely, that some Vice President voted against the administration despite being a member of the same party as the President.