This is a great question. I think it's better to not focus solely on the tree of life but the entire garden, as this gives a better picture.
God could have put a flaming sword around the tree of life and let Adam remain in the garden. But he chose to expel man from the entire garden and set the flaming sword and the two cherubs to guard the entrance to the garden in order to prevent access to the tree of life. Why? The garden somehow was associated to the tree of life and could not be separated from it.
So what did the garden have in common with the tree of life?
I believe the garden was filled with life. "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." (Gen 2.9)
Thus the garden was fertile or brimming with life, and out of this fertile life sprang the tree of life, which is the apex of life giving fruit.
Thus we have the beginning of a holiness code, in which God says "this area is for life, and this for death". Man was made not from the fertile ground of the garden, but from ordinary ground, and only put into the garden after God breathed the "breath of life" into man (Gen 2.7) so that man became living nefesh. It's interesting that nefesh and hayyah are two words that both have the sense of living and the parallel the promised two deaths if the tree of knowledge was eaten (מ֥וֹת תָּמֽוּת or mot tamut in Gen. 2.17).
Only after the man was imbued with life by God could he enter this garden. This suggests that there were already requirements for entrance, and one of man's jobs was to "guard" the garden (shamar in Gen 2.15, which means both "take care of" and "guard" as does the Elizabethan word "keep".)
At the fall, man was defiled and dead (2.17), and thus could not step on the holy ground and so was sent back into the (now) cursed ground, to work it, in contrast with the holy ground that he was cultivating before the fall.
Thus this is the beginning of the various holiness codes as well as temple layout. You have:
blessed garden cursed ground
| Holy |
| Most holy | |
| tree of life | cherub+fire | outside
Many temples in the ancient near east followed this pattern, of
a holy place, and an inner "Most Holy" place[1]. At the entrance to the
holy place would be a sacrificial altar of fire. Cherubic imagery would be used throughout.
Thus you have this concept of holiness, or being set apart for God, in which the world is divided into life and death, clean and unclean. Therefore the main source of uncleanness is touching a dead body. When God said "you shall surely die", man became dead, and thus could not remain in the holy area.
Later on, God would say that all of the promised land was holy land, and all of Israel was a holy nation, and thus everyone in Israel had to avoid touching a dead body, and a number of related ritual purity requirements were instituted and grew out of this basic conception.
Moreover the garden was guarded by the flaming sword and the two cherubim, symbolic of divine fire.
Later on, the temple area had to be ritually purified by sin offerings consisting of the blood of the scapegoat for the nation, as well as bulls for the priesthood, male goats for the priest, and female goats for ordinary people, all of which were put on the altar and burned (although only the purification sacrifices whose blood was sprinkled on the holy of holies were entirely burned). There was also the red heifer which was fully burned, and the ashes of the red heifer mixed with living water purified the unclean (Numbers 19).
So out of this basic picture of the divine fire at the gate came the sacrificial system for restoring holiness (there were other types of offerings that I think are less relevant). Note that the brazen altar was lit with divine fire from Heaven (Lev 9.24) and this fire was never to be put out but was to be kept burning continuously (Lev 6.12). When Solomon dedicated his Temple, divine fire again fell from Heaven to light the altar (2 Chron 7.1).
After the fall, God clothed man with animal skins, which required the sacrifice of an animal, and then Adam wore its skin, and was driven out. Thus there were two identical animals, the one sacrificed and the one driven out. This ritual supposedly cleansed the garden from the defilement of Adam's sin, and so similar purification practice was used in the scapegoat sacrifice (Lev 16) and the pair of two birds (Lev 14.3-7), both involving two identical animals, with one sacrificed and the other sent out.
So rather than just looking at the tree of life (the most holy area), look at the entire layout and see within it the same general principles that form much of the Levitical system that enforce the separation of life from death.
Thus the answer to the overall question would be that life was to be separated from death. Holy things containing life were to be separated from cursed things that contained or touched death (such as man and his decedents). Eden was holy, and the tree of life, as something that gave endless life, was the most holy and thus most in need of separation from the world of death.
[1] https://emp.byui.edu/satterfieldb/rel390r/Reading%20Assignments/What%20is%20a%20Temple%20JML.pdf
