Suppose Trump dies and Pence becomes President, leaving the Vice Presidency vacant. Before a new Vice President is installed, the Senate ties a vote. Who breaks the tie?
For a resolution to pass, the "Yeas" must be greater than the "Nays". If there is a tie, then the Vice President may cast a vote to break the tie. If there is no vice president and the vote is tied then the resolution doesn't pass.
It is rare for this to have happened, which is surprising since it was common practice, until the 25th amendment, for the President to leave the VP position open. There is the story of how Texas was admitted to the Union by one vote 27-25, since a 26-26 tie would have resulted in Texas not entering, (Despite the President's support) as there was no Vice President to break the tie.