What would happen to somebody voting in opposition to a proposition at LDS General Conference?

I just wanted to chime in here though the question has been answered fairly well.

For awhile there was a group of LDS that were opposed to one prophet in particular. Every conference they would hold up signs denouncing him and those able to make it into the conference hall would vote against him.

The reason I bring this up is to show how this is dealt with at that level. There are actually observers throughout the conference hall and anyone voting in dissent is noted, and given a chance to state their objections to the proper authority then. They are not directed to take it up elsewhere or shuffled off.

The dissenters in the group I referred to were - despite it being the same objection every 6 months, and a boundless one at that - given the chance to voice their opinion, and were never barred from the proceedings of general conference.

During the October 1980 General Conference there were some people who opposed the church president. They were told they could meet with one of the apostles after the session. You can view it, it is on the LDS church website where the General Conference videos are posted.

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:

D&C 28:13

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.

This originates from a revelation given to Joseph Smith, recorded in Doctrine and Covenants 26, and is known as the law of common consent. Note that this is different from congregationalism in that in common consent, a decision is already made through prayer and by inspiration, and the decisions are not made by democratic voting, per-se. Those affected by the decision or change then sustain it, not necessarily vote on what should or should not be.

Sustaining leaders is a regular occurrence in the church, even in local meetings. Raising the right hand may be understood, in some way, as a pledge or covenant to help and support whoever is being proposed.

If somebody opposes, it is generally done because of some first-hand knowledge the opposer has of the person in question. In local settings, that person will be asked why after the meeting (usually the reason must be for some grave sin, or unworthiness to serve in the position, which leaders usually know about anyway as they may help the person through the repentance process).

In general conference, there are about 21,000 people in a single session in the Conference Center. The conducting authority will generally conclude with "Insofar as I have been able to observe, the voting in the conference center has been unanimous in the affirmative."

If there is an opposition in general conference, the person is directed to speak with their stake president who takes the matter, if it merits attention, to higher authorities.

Update: I was in attendance when this happened today, on Saturday, April 4, 2015, during the afternoon session of General Conference. Some attendees opposed and the speaker simply remarked, "Thank you, the vote has been noted." Instead of saying that voting was unanimous at the end, he said, "The voting has been noted. We invite those who opposed any of the proposals to contact their stake presidents." ^[Video]