Must We Vote The Way The Brethren Tell Us To Vote?

Consider these quotes showing that the Church should not be involved in things "rendered unto Caesar" -

"That this Church, while offering advice for the welfare of its members in all conditions of life, does not claim or exercise the right to interfere with citizens in the free exercise of social or political rights and privileges. "
(James R. Clark, comp., Messages of the First Presidency of The Church of Jesus Christ of Latter-day Saints, 6 vols. Salt Lake City: Bookcraft, 1965-75), 3: 185.)
In other words, the Church shouldn't ask its members to help pass laws objectively harming the free exercise of social or political rights and privileges of others.

"The Church believes that political and religious influence should be kept apart. It does not in any way desire to encroach upon the rights or privileges of the State. It only rises to defend encroachments upon man's right of conscience."
(John A. Widtsoe, Program of The Church of Jesus Christ of Latter-day Saints [Salt Lake City: The Church of Jesus Christ of Latter-day Saints, 1937], 100 - 101.)
The Church shouldn't be asking us to vote a certain way when the issue at hand in no way infringes upon the Church or the free exercise of social or political rights and privileges of others.

Bruce R. McConkie, in his Doctrinal New Testament Commentary, gives us additional doctrinal support to oppose the banning of Civil Same-Sex Marriage. He discusses -

Matthew 22:15-22, Mark 12:13-17 and Luke 20:22-26. He states -

"How better could the Master Teacher proclaim his doctrine of separation of church and state than he did here in avoiding the trap of the Pharisees and Herodians?
These scheming and hypocritical "spies" offered Jesus these two alternatives as possible answers to their diabolically clever question: (1) Say, 'Yes, pay the hated poll tax to Rome as the law already requires,' which answer would cause the Pharisees to inflame the people against him. Or: (2) Say, as the sect of Zealots taught, 'No, Israel is a theocracy; God only is her King; pay no tribute to a foreign power,' in which event the Herodians would have delivered him "unto the power and authority of the governor," charging him with sedition and rebellion.
But from his lips instead came the decree: 'Render unto God and Caesar their own.' That is, in this present world where wicked men will not repent and come unto the fulness of the Lord's perfect order of government, there must be two separate powers - ecclesiastical and civil - the one supreme in spiritual matters, the other in temporal. Neither power can dictate to the other. And men are subject to them both."
(Bruce R. McConkie, Doctrinal New Testament Commentary, 3 vols.,[Salt Lake City: Bookcraft], 1:600,601.)

The civil and legal rights and benefits pertaining to marriage are clearly the domain of government (Caesar). We know that marriage has a legal/secular component because even clergy state that they perform marriages "by the power vested in me by the state of...I pronounce you husband and wife". We also know that people can't go to their clergyman to obtain a legal divorce. These aspects of marriage are clearly the domain of Caesar thereby refuting the idea that marriage is solely the domain of the church. Religious belief and doctrine have no right to infringe on the rights and liberties of others. D&C 134:4 confirms this, as does the following quote -

"Religious freedom does not imply nor provide license to infringe or impose upon the rights and liberties of others."
(L. G. Otten and C. M. Caldwell, Sacred Truths of the Doctrine and Covenants [Salt Lake City: Deseret Book Co., 1982-1983], 2: 375.)

Consider the following quotes showing that members can vote against the wishes of the Brethren and still be considered worthy Saints -

"Some of my friends begged me to come out and appeal to the people individually, to ask every Latter-day Saint to vote to maintain the Eighteenth Amendment. I believe men that have lived the gospel just as well as I have ever lived it, many of them, were conscientious in voting for repeal."
(Heber J. Grant, GOSPEL STANDARDS - Selections from the Sermons and Writings of Heber J. Grant, Compiled by G. Homer Durham, Ch.9, Pg. 144)

"As a public official in my young manhood, I was given some wise counsel by a Church leader. He said: "The only action we will ever ask you to take is to vote for that which in your heart you feel is right. We would rather many times over that you would make a mistake doing that which you felt was right, than to vote for a policy sake."
(Harold B. Lee, Ye Are the Light of the World: Selected Sermons and Writings of Harold B. Lee [Salt Lake City: Deseret Book, 1974])

As the above quotes show, we can vote differently than the Brethren wish we would and still be just as righteous. You are to "vote for that which you felt was right, (rather many times over) than to vote for a policy sake." This is especially true when the thing being proposed is contrary to the scriptures.

Only if the prophet declares that the Lord revealed to him that we are to vote a certain way and we, the members of the Church, support that declaration via Common Consent does it become binding upon us. In such a case, voting against it would be considered a sin.

Consider the following quotes which show that anything official and/or binding must be ratified by Common Consent -

"I am afraid, however, that this is not as generally accepted or followed to-day as it ought to be. Some of the brethren have been willing to submit to the inference that what they have said was pronounced under the influence of the inspiration of the Lord and that it therefore is the will of the Lord. I do not doubt that the brethren have often spoken under inspiration and given new emphasis --perhaps even a new explanation or interpretation--of Church doctrine, but that does not become binding upon the Church unless and until it is submitted to the scrutiny of the rest of the brethren and later to the vote of the people."
(Hugh B. Brown, The Abundant Life [Salt Lake City: Bookcraft, 1965])

(I)t is not the prerogative of the people to vote whether or not a particular truth or revelation is valid. Opportunity may be given to the members, however, to indicate whether or not they favor the publication of a particular revelation with the official canon (scriptures) of the Church; they may also be given the opportunity to bind themselves by covenant to follow the instructions contained in the revelation.
(Daniel H. Ludlow, A Companion to Your Study of the New Testament: The Four Gospels [Salt Lake City: Deseret Book Co., 1982], 43 - 44.)

"In no sense can the Church be called autocratic. No one, from the President down, can dictate to the Church. All must be done in harmony with gospel principles, and by common consent. Even new revelations from the Lord are presented to the people for acceptance as part of the doctrine of the Church."
(John A. Widtsoe, Evidences and Reconciliations [Salt Lake City: Improvement Era], 281.)

"Members do not nominate persons to office, but are asked to give their sustaining vote to decisions of presiding councils by raising their right hand, and anyone may give an opposing vote in the same way. This procedure is also followed in accepting important revelations and scriptural additions."
(Encyclopedia of Mormonism, 1-4 vols., edited by Daniel H. Ludlow (New York: Macmillan, 1992), 298.)

This sentiment is echoed here by FAIR, the premier LDS organization / website dealing with LDS apologetics.

The prophets have also condemned the idea that we blindly follow them -

"I am more afraid that this people have so much confidence in their leaders that they will not inquire for themselves of God whether they are led by him. I am fearful they settle down in a state of blind self-security, trusting their eternal destiny in the hands of their leaders with a reckless confidence that in itself would thwart the purposes of God in their salvation, and weaken that influence they could give to their leaders, did they know for themselves, by the revelations of Jesus, that they are led in the right way. Let every man and woman know, by the whispering of the Spirit of God to themselves, whether their leaders are walking in the path the Lord dictates, or not."
(Brigham Young, Discourses of Brigham Young, selected and arranged by John A. Widtsoe [Salt Lake City: Deseret Book Co., 1954], 135.)

Finally, consider these words of Joseph Smith on this -

"We have heard men who hold the priesthood remark that they would do anything they were told to do by those who preside over them [even] if they knew it was wrong; but such obedience as this is worse than folly to us; it is slavery in the extreme; and the man who would thus willingly degrade himself, should not claim a rank among intelligent beings, until he turns from his folly. A man of God would despise the idea. Others, in the extreme exercise of their almighty authority have taught that such obedience was necessary, and that no matter what the saints were told do by their presidents they should do it without any questions. When Elders of Israel will so far indulge in these extreme notions of obedience as to teach them to the people, it is generally because they have it in their hearts to do wrong themselves."
(Joseph Smith, Millennial Star, Vol 14, Number 38, pages 593-595)

The bottom line is that we, the members of the Church, are in no way required by doctrine to vote in a certain way, even if asked to by the prophet. This is especially true, as stated above, when the thing being proposed is contrary to the scriptures. It is also true unless we, as a Church, vote via Common Consent to sustain the prophet's request. Without that sustaining vote, we are to "study it out in our minds", search the scriptures, and pray to determine which path is in keeping with the Father's will. We are to rely on the strength of the Lord rather than on the "arm of flesh".