JOEL - The "tree of life" was basically the representation of immortality and eternal life. There was nothing to stop Adam and Eve from eating from this tree; in fact they probably did eat from it while in the garden.
When they did eat nothing would have happened to them because they were already immortal. As such they would have lived forever in the Garden if they would not have partaken of the tree of knowledge of good and evil (2 Ne. 2:22-23). It was only after they partook of the forbidden fruit and became mortal , that they could not be allowed to eat from the tree of life; for if they did it would have caused them to live forever in their fallen state and frustrate the plans of God.
Alma explains it best in the Book of Mormon:

"21. What does the scripture mean, which saith that God placed cherubim and a flaming sword on the east of the garden of Eden, lest our first parents should enter and partake of the fruit of the tree of life, and live forever? And thus we see that there was no possible chance that they should live forever.
22. Now Alma said unto him: This is the thing which I was about to explain. Now we see that Adam did fall by the partaking of the forbidden fruit, according to the word of God; and thus we see, that by his fall, all mankind became a lost and fallen people.
23. And now behold, I say unto you that if it had been possible for Adam to have partaken of the fruit of the tree of life at that time, there would have been no death, and the word would have been void, making God a liar, for he said: If thou eat thou shalt surely die." (Alma 12:21-23)

and

"5. For behold, if Adam had put forth his hand immediately, and partaken of the tree of life, he would have lived forever, according to the word of God, having no space for repentance; yea, and also the word of God would have been void, and the great plan of salvation would have been frustrated.
6. But behold, it was appointed unto man to die—therefore, as they were cut off from the tree of life they should be cut off from the face of the earth—and man became lost forever, yea, they became fallen man. (Alma 42:5-6)

The protection of the tree (verse 21) was for a two fold purpose: to bring about a guarantee of death as God had proclaimed, and to allow a probationary time for mankind to repent and serve God. This probationary period was to allow mankind to prepare themselves to meet God at the judgement bar, being cleansed from their sins through the atonement.