Interesting question.

In the case of a Vice Presidential vacancy, the President Pro Tempore of the United States Senate assumes the responsibilities and powers of the Vice President in his duty as President of the Senate but cannot break a tie. If the result of a vote on a Senate action results in an unbreakable tie, the action fails to pass.

It doesn't seem to have happened very often but one notable case did occur when the Senate was tied and for a brief period there was no sitting Vice President. It resulted in a brief constitutional crisis of sorts when control of the Senate hung in the balance. "In 1881, the Senate stood at 37-37, with 2 Independents: one tended to vote with the Democrats and the other with the Republicans. When Vice President Chester Arthur became president after the assassination of President Garfield, the Vice Presidency was left vacant, leaving no one to cast a tie-breaking vote in the Senate. The two parties came to an agreement where the Republicans controlled the committees and the Democrats managed the patronage." Quote taken from here:.

Edited: A ton of stuff. For clarity. And mistakes.