"The tree of life" of Revelation?
When John uses the term "the tree of life" in Revelation 22:2 he is referring to something familiar to both him and his audience. (In other words, he was speaking within the semantic range of the term.) Given that the only "tree of life" that appears in Scripture is in Genesis 3:22, and that John's Apocalypse is replete with Biblical allusions, it is a virtual certainty that this is what he was referring to. Otherwise we would expect him to say "Then I saw something like the tree of life . . ." (cf. Eze. 1:27, Rev. 15:2.)
It is worth noting that this is what John saw. There are many different interpretations of what his vision signifies.
God's reason for not wanting Adam to live forever?
Since we're on the hermeneutics site, I need to stick to exegesis and not theology for this answer. There are several clues in the context of Genesis 3:22 that may help to answer this question:
Three verses prior, God had determined that man would "return to the ground . . . For you are dust, and to dust you shall return." Man was thus sentenced to death. This is consistent with the warning of Gen. 2:17 that in the day they ate from the tree they would surely die. If God then allowed them to eat from the tree of life and live forever that would thwart His justice in 3:19.
In the next verse we read that God "drove the man out; and at the east of the garden of Eden He stationed the cherubim and the flaming sword". Man was now doomed to live out his days away from the paradise where they once walked with God. (cf. 3:8) If God had allowed them to eat from the tree of life and live forever, it would be outside of the paradise where they once walked with God.
Immediately after the sentence of 3:19, and immediately before the judgment of 3:22, we see God Himself clothing the man and his wife. This indicates that despite their sin and death sentence, God is not done with them. Many interpreters see in this an early allusion God's plan for their redemption to eternal life with Him despite the certainty of their physical death. If that is correct, then if God had allowed them to eat from the tree of life and live forever, that would conflict with His plan for redeeming them from certain death to eternal life with Him.
Why would God then let people live forever after the final judgment?
Having already explained why God didn't want Adam to live forever in Gen. 3:22, the only remaining question is why God would want / allow people to live forever after the final judgment (following their physical death.)
As mentioned, Gen. 3:21 indicates God was not done with them, despite their death sentence. If this glimmer of hope points to a future redemption despite the certainty of their physical death, that redemption would be totally consistent with the idea of eternal life presented in Matt. 25:46.
The eternal death would simply be a continuation of the judgment experienced here on earth for those who were not redeemed.