There are several ways to look at this question.
A. Does the LDS Church believe the Sacrament (the "sacrament of communion" or the Eucharist) when administered by another church or Christian tradition is authoritative?
No. We do not.
We believe that "a man must be called of God, by prophecy, and by the laying on of hands by those who are in authority, to preach the Gospel and administer in the ordinances thereof." (LDS AoF #5). Without that authority, the Sacrament is not acceptable before the Lord (Jer 6:18-20).1 This belief is supported by the Apostle Paul:
Let every soul be subject unto the higher powers. For there is no power but of God: the powers that be are ordained of God. Whosoever therefore resisteth the power, resisteth the ordinance of God: and they that resist shall receive to themselves damnation.2 (Rom 13:1-2, see also Heb 5:1-4)
We believe John the Baptist had the proper authority by lineage to be a priest within the Aaronic Priesthood:
Son of Zacharias and Elisabeth, being of priestly descent through both parents. This lineage was essential, since John was the embodiment of the law of Moses, designed to prepare the way for the Messiah and make ready a people to receive Him. He was the outstanding bearer of the Aaronic Priesthood in all history and was entrusted with its most noble mission. (Source, see also D&C 84:28)
Therefore, we do not accept the authenticity of ordinances by anyone not ordained by those who have the authority to the Aaronic or Melchizedek priesthoods.
But that's very different than...
B. Does the LDS Church teach its members to refuse participation in the Sacrament (the "sacrament of communion" or the Eucharist) when administered by another church or Christioan tradition?
There is no Church teaching that I am aware of, or can find in rapid research, that teaches that a member of The Church of Jesus Christ of Latter-day Saints should not accept a sacramental offering (from the LDS perspective) administered by an officiator of another church.
The idea that we would act with such incivility and rudness is (or, at least, should be) anathema to us. We delight in every opportunity to teach others about our faith, but we should never persecute another for their faith. The Lord taught:
And whosoever shall offend one of these little ones that believe in me, it is better for him that a millstone were hanged about his neck, and he were cast into the sea. (Mark 9:42)
And Jesus said unto him, Forbid him not: for he that is not against us is for us. (Luke 9:50)
Regrettably, those ideals are applicable to the Church and its membership generally and not individually. We do not believe that people are perfect or saved simply because they join the Church, and there are members who behave less than admirably.
Nevertheless, there are reasons why it would be appropriate and valuable for an LDS member to support the solidarity of a group in crisis. What value could there be in creating ill will, dissent, or anger when the next moment a battle will begin or the group must steel itself before rescue? The Lord taught that we should agree with our adversaries quickly, lest the resulting disunity3 do more harm than good (Matt 5:25) And He ultimately taught that we should love even our enemies, blessing those who hate our Church, and praying for those who would persecute us. (Matt 5:44)
There is nothing that I know of, nor anything that I believe, that would suggest an LDS member should not participate in the Sacrament as provided by another officiator in the circumstances you suggest. On the contrary, the teachings of Jesus Christ are that we should be supportive and consider the needs of the many before our own selfish desires,4 always acting within the law that Christ has taught us,5 that our good example may grant us opportunity to teach Christ all the more. (1 Cor 9:19-22)
(While I have not been in a crisis situation, I have been invited by friends to attend their church services, where I politely participated in their ordinances and suspect the Lord was delighted with their effort to grow closer to Him and my willingness to let a sleeping dog lie for a greater good. Thus, I do not believe I was in error in any way and the courtesy was returned when they visited my church.)
And though it is true that we do not accept the officiator's authority, we should never act to bring anger, hate, or distrust among a group that must depend on one another (or ever, frankly).
I am grateful for a simple quote from Victor Hugo.
Certain thoughts are prayers. There are moments when, whatever be the attitude of the body, the soul is on its knees.
There really are no atheists in foxholes. When the pressure's on, all men seek for salvation.
1 That verse from Jeremiah, from the perspective of LDS theology, supports the idea that without proper authority the ordinances performed are not acceptable to the Lord. This is due in part to D&C 131:7 which teaches that when any priesthood holder is wicked, "behold, the heavens withdraw themselves; the Spirit of the Lord is grieved; and when it is withdrawn, Amen to the priesthood or the authority of that man." Said another way, to the Lord it doesn't matter if you had the priesthood and are in sin, or didn't have the priesthood at all. The consequence is the same: the ordinance goes unrecognized.
2 I'm obviously viewing this verse through the lenses of LDS theology. Whether or not ours is the true authority of God is a discussion not within the scope of this answer.
3 I consider being thrown into prison an obvious sign of disunity and to be more harmful than good both theologically and physically. I'm just sayin'.
4 Yup, I'm a Star Trek fan.
5 As mentioned by others, we wouldn't drink the wine, but would ask for an alternative.