Does the LDS discuss the issue of LDS children "playing Sacrament"?

To answer the main question about "playing sacrament," as the play is by children, with children, and without authority, there really isn't much of a problem—especially if it is done with a sense of reverence.

In regards to unbaptized children actually partaking of the sacrament, while I don't disagree with the previous answers, I think they really miss the heart of the answer. Going into the way-back machine, we read Genesis 2:8–9,15–17:

8 ¶And the Lord God planted a garden eastward in Eden; and there he put the man whom he had formed.

9 And out of the ground made the Lord God to grow every tree that is pleasant to the sight, and good for food; the tree of life also in the midst of the garden, and the tree of knowledge of good and evil.

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15 And the Lord God took the man, and put him into the garden of Eden to dress it and to keep it.

16 And the Lord God commanded the man, saying, Of every tree of the garden thou mayest freely eat:

17 But of the tree of the knowledge of good and evil, thou shalt not eat of it: for in the day that thou eatest thereof thou shalt surely die.

From this passage, we learn that the Tree of Life was in the garden, that Adam and Eve had access to the Tree of Life, and that it was one of the trees from which they (in their current, innocent state) could freely eat.

In the next chapter, after loosing their innocence and partaking of the fruit of the Tree of Knowledge of Good and Evil, we read Genesis 3:22–24:

22 ¶And the Lord God said, Behold, the man is become as one of us, to know good and evil: and now, lest he put forth his hand, and take also of the tree of life, and eat, and live for ever:

23 Therefore the Lord God sent him forth from the garden of Eden, to till the ground from whence he was taken.

24 So he drove out the man; and he placed at the east of the garden of Eden Cherubims, and a flaming sword which turned every way, to keep the way of the tree of life.

Therefore, it was acceptable for Adam and Eve to partake of the Tree of Life until they lost their innocence. From a vision in the Book of Mormon, recorded in 1 Nephi 11, there is an equivalence given between the Christ and the Tree of Life. The LDS Sacrament or Holy Eucharist is also a representation of the Christ.

It would then seem reasonable then that as long as someone maintains their innocence (cf. Doc&Cov 29:46–47), they may partake of the emblems of Christ freely.

As far as 'playing' sacrament, I don't know that I've ever heard of that, or whether it was an issue.

As to whether children should partake of the sacrament at all, there is an article in a recent edition of the Ensign, the LDS's official monthly magazine, containing one member's thoughts and reflections on this issue. Although it is not directly considered LDS teaching, since it was published in the Ensign, it can be considered at least a tacit endors*m*nt that could be helpful in answering this question.

Little Children and the Sacrament.

Have you ever wondered why we allow unbaptized children to partake of the sacrament? ...

I believe this because I believe that when Jesus Christ says “all,” He means all. And when He speaks to a multitude, He doesn’t exclude anyone. ...

We help our little ones partake of the sacrament so they can remember their Savior and keep His peace—a peace unlike anything the world can offer.

I have been unable to find an answer that answers your question exactly but there are some teachings that may help you come to a decision yourself. My answer comes from Answers to Gospel Questions volume 2 (question #18) and 5(question #17) by Joseph Fielding Smith.

From AtGQ Vol 2 #18

It may be true that some of the smaller children who attend the ... sacrament meetings..are too young to realize the significance of this sacred ordinance, but what injury can come to them or to the ordinance if they are permitted to partake? We know that the feelings of our Savior were very tender toward little children.

From AtGQ Vol 5 #17

Contrary to the expressed opinion of some who think that the partaking of the sacrament at such an early age "detracts from its meaning," we sincerely proclaim that the permitting of small children to partake when it is explained to them acts upon their minds exactly to the contrary. Perhaps some of us fail to comprehend the depth of a small child's mind. Surely little children in the Primary class can understand that the sacrament is a sacred ordinance when they are carefully taught. ... No one will doubt that these little ones were so deeply impressed that they remembered this glorious occasion all the days of their lives. Perhaps we at times are too hasty in our judgment in relation to the capacity of small children to understand.

Matthew 19:14-15 says

14. But Jesus said, Suffer little children, and forbid them not, to come unto me: for of such is the kingdom of heaven.
15. And he laid his hands on them, and departed thence.

And Matthew 18:3-6 says

3. And said, Verily I say unto you, Except ye be converted, and become as little children, ye shall not enter into the kingdom of heaven.
4. Whosoever therefore shall humble himself as this little child, the same is greatest in the kingdom of heaven.
5. Whosoever therefore shall humble himself as this little child, the same is greatest in the kingdom of heaven.
6. But whoso shall offend one of these little ones which believe in me, it were better for him that a millstone were hanged about his neck, and that he were drowned in the depth of the sea.

We know from D&C 29:46-47 that little children are saved if they die at a young age and that Satan has no power over them to tempt them. D&C 68:25-28 mentions that parents are commanded to teach their children to understand the principles of the gospel from their earliest childhood so they will know why these ordinances and principles are essential parts of the gospel of Jesus Christ. As long as you can turn their play into a teaching opportunities there doesn't seem to be any reason they can't "play sacrament".