Yes, the Tree of Life is actually a common motif for seemingly many cultures -- not just the Near East. Its symbol and presence in mythology transcends geographical boundaries. Just conducting a basic google search, Wikipedia cites all the many cultures that have portrayed a "Tree of Life" in their mythical/philosophical traditions: Tree of life (Wikipedia). As you can see, the cultures range from Ancient Iran to China to Europe to Mesoamerica. The existence of trees all around the world and their ability to grow, blossom, and provide fruit likely make the "Tree of Life" a pan-human archetype, especially in regards to concepts concerning "eternal life" since trees play a prominent role in the circle of life.
In regards to specifically Near Eastern motifs, I would like to provide another resounding "yes." In fact, I'm not sure if you're aware, but it is also present in Judeo-Christian traditions. Everyone remembers the "Tree of Knowledge of Good and Evil" but many forget the other tree in the Garden of Eden -- "The Tree of Life." Genesis 2:9 and 3:22-24 mention the "Tree of Life" in the Garden of Eden story, which most certainly concerns the creation/purpose of mankind and can easily be linked to concepts of "eternal life." Many biblical scholars also propose that the chapter 22 of the Book of Revelation makes reference to it again since it addresses the new garden of paradise and only the righteous will be able to partake in such blessings.
Additionally, Proverbs 11:30 & 13:12 directly reference the Tree of Life, but in Proverbs it generally symbolizes wisdom. In the non-cannonical Book of Enoch, the names of the rigetious will be allowed to receive and taste the fruits of the "Tree of Life." Though non-cannonical, it shows that the idea was alive and kicking in antiquity. Saint Isaac the Syrian (from Nineveh) circa 613 – 700 expands upon this matter and elucidates: "Paradise is the love of God, in which the bliss of all the beatitudes is contained," and that "the tree of life is the love of God." (Homily 72). This seems to correspond to the LDS interpretation that you have listed above.
But assuming you're interested in "Tree of Life" stories that would predate or concurrent with the lives of the LDS prophet Lehi (600 BCE), I would take a special look at The Epic of Gilgamesh. It doesn't cite a "Tree of Life" directly, but it focuses on Gilgamesh's quest for eternal life, and the elixir is a plant (which is unfortunately consumed by a snake). Not a direct relation, but the commonality of a plant providing immortality and a snake disrupting such plans seems pretty interesting.
Ancient Egyptian texts also center around the Acacia Nilotica, which is known as the "Tree of Life." The fruit of this tree would provide the partaker not only with eternal life, but with a plan, i.e. map, on how to get that point: The Tree of Life. This may be of peculiar interest to you since the Book of Mormon mentions that Lehi was learned in the language of the Egyptians.
There are also a series of panels in Nimrod (ancient Assyria), which depict a "Tree of Life" motif, but scholars have not reached a consensus on what it represents exactly, but you can find images of it here.
Your question did not ask about "Tree of Life" motifs in the New World, but since you are broaching a Mormon question, I figured you would have an interest in this region as well. Take a first look at the wikipedia article I linked to see just how many pre-columbian cultures had Trees of Life in their iconography. But I would also recommend taking a gander at this source: Miller, Mary; Karl Taube (1993). The Gods and Symbols of Ancient Mexico and the Maya. London: Thames and Hudson. The trees connect the realms of the living with the realms of the heavens -- an axis mundi, if you will.
Clearly, there are many representations of it. I hope that helps! Thanks for asking the question -- it was an interesting one to answer.