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First, if we take the tree to symbolize a pledge to Adam the immortality and all that life required, based on continued obedience to all the commands of God, then the tree was a pledge to Adam that so long as he did not eat from the forbidden tree he would have lived forever. This is reasonable.
Second, by sinning Adam lost that immortality, therefore the pledge must necessarily be removed and access to it be forbidden.
The actual death of Adam for sinning was not due to anything inherent in the fruit of the tree physically, but in his internal disobedience. Similarly, eating from the physical tree of life could not have redeemed his soul to life, or his body, as there is no such 'magical fruit' that can forgive the consequences of sin, or kill. Rather when God says 'they must not eat of it', he is referring to the eternal life as a reward which presumes full obedience to all his laws. This Adam must no longer physically do, for the act would betray its symbolic meaning. God physically made it impossible to eat from the tree with flaming swords reminding him of his cursed state, in order to maintain the symbolic messages both trees conveyed. In other words, Adam could not physically eat from the physical tree because he could not internally obey all God's commands in his depraved sinful state and ...live forever in so doing. The tree needed to be replaced by a new tree of life, i.e. Christ dying on a tree, whereby we may eat and live forever.