I know that the VP breaks a tie in the Senate.
What happens if there is a tie vote in the House of Representatives?
Standardly, whoever is presiding at the time of the vote does not vote. If the vote is tied, then that person casts the deciding vote. In any case a tie is not an affirmative vote so the motion does not carry.
But unlike the Senate, the presiding officer of the House is (generally) one of its members (I don’t think there’s an actual rule to that effect, but it’s always been the case). Doesn’t that mean, in effect, that the presiding officer’s side loses all tie votes?
I believe the Speaker’s refraining from voting is not binding. He or she can vote if they wish and would presumably do so if the vote would otherwise be tied. Or in some cases it they want to cast a vote because of the significance of the issue being voted on. If you saw the movie Lincoln, there was a scene where Speaker of the House Schuyler Colfax interrupted the clerk who had skipped over him during the roll call to say he wanted to vote on the 13th Amendment.
Traditionally, the Speaker of the House does not vote. There is no rule about this, just history. He or she can (and sometimes does) so a tie could be broken. If the vote is going to be lost anyway, then there is no point since a tie does not pass.
Each party has members whose job it is to count votes, known as whips. There are no surprise votes in Congress, at least in modern times. No vote would be allowed to take place if the outcome weren’t a known victory, just as happened with the Health Care Bill be pulled.
True the outcome of the bill is known in advance of the vote. But sometimes the leadership will let bills go down to defeat by actual votes, just so they can use the defeated bill as a campaign issue against the people who voted it down.
Steve Bannon apparently wanted this to happen with the health care bill, so they could compile an enemies list of the Republicans who voted against it. Cooler heads prevailed.