How does the following line from Leo Tolstoy’s The Death of Ivan Ilyich contribute to the plot at the end of the novella? "Death is finished, he said to himself. It is no more!"

Obviously, Ivan is not claiming he will not physically die. This quote refers to his spiritual regeneration, which only came through the process of death waking him up from the emptiness of his loveless, materialistic life. These lines show that Ivan has made his peace with death after finally realizing that the meaning of a good, proper life is loving others, rather than getting as high up on the social ladder as possible. Now that he realizes the illusory nature of the material world, which was his entire world before his final illness, he is no longer frightened of dying.
Also, these lines are evocative of certain lines from the New Testament, such as Jesus's dying words, "It is finished" or Paul of Tarsus's famous claim in 1 Corinthians that death has been swallowed up in the victory of Christ. Like Jesus, Ivan has gone through a painful death to achieve a great spiritual victory.