While it may appear similar to a vote, (and is often referred to by that term,) these sustaining votes are not a democratic election, but rather an opportunity for members of the church to show their agreement or opposition to the choices that have been made with regard to leadership positions. These sustaining votes take place not only at General Conference but also at local meetings when people are called to serve in some position in the congregation.
The principle is known as "common consent", as described in the Doctrine and Covenants:
For all things must be done in order, and by common consent in the church, by the prayer of faith.
Latter-Day Saints generally accept as a point of faith that the church is run by inspiration and could not operate without it, as declared by President Wilford Woodruff in 1891:
It matters not who lives or who dies, or who is called to lead this Church, they have got to lead it by the inspiration of Almighty God. If they do not do it that way, they cannot do it at all.
Because of this, a sustaining vote, when it is called for, is usually answered unanimously in the affirmative. However, this is not regarded as a formality. Church leaders are still human, and it's recognized that they can make mistakes, even when attempting to act in all sincerity and soberness.
Occasionally when a person is called to serve in some position, if some of the members of the congregation are aware that they are not worthy or should not serve in this position for whatever reason, they are allowed and asked to show their objections here. If this happens in a local meeting, the person making an objection would talk to the Bishop in private later, and the member who had been called to serve would not be set apart for their calling until things had been resolved.
If it were to happen during General Conference, I don't know how it would be dealt with, as I've never observed it during my lifetime. I have heard that before I was born, there were a few instances where anti-Mormon protestors managed to gain admission to General Conference and used this time as an opportunity to shout "NO!" or chant slogans in an attempt to disrupt the proceedings. As I understand it, their agitations were dealt with by simply ignoring them.