Question: When a person dies does it mean that he has completed his mission in this world?
Answer: No. There are also wicked people who do the exact opposite of their missions and they nonetheless die. The essence is therefore for a person to utilize every moment of his life to do good. See in the Siddur of Maran Ha-Rav Kook Volume 2, p. 364 on the Davening for Yom Kippur.
Once during the Yom Ha-Atzmaut or Yom Yerushalayim celebration in Yeshivat Mercaz Ha-Rav, they also celebrated the 80th birthday of Rabbenu Ha-Rav Tzvi Yehudah. Everyone praised Rabbenu and all of his life's achievements. Ha-Gaon Ha-Rav Shlomo Yosef Zevin stood up and said: I do not agree with all of these praises. He related that people once came to the Kotzker Rebbe with a sick child and requested that he prayer for their precious and wonderful child, and they recounted all of his sterling qualities. The Kotzker responded that the child had not done a thing. Everyone was shocked! Instead of arousing merit for the child, he denounced him. The child nonetheless recovered. The Kotzker Rebbe said that the Gemara in Kiddushin (31b) tells that Rebbe Tarfon's mother came to the Beit Midrash and said: Pray for my son who is a great Tzadik. The Rabbis asked: In what way is he a great Tzadik? She said: I once lost my shoe and he placed his hand under my foot the entire way home. They said to her: This is nothing! Even if he did 100 times this, he still would not fulfill half of the Mitzvah of honoring one's parent. The Kotzker asked: Why did the Rabbis belittle Rabbi Tarfon? He explained that what Rabbi Tarfon did was indeed great, but they did not want to ascribe it too much importance for fear that it would mean that he had completed his role in the world. Our Sages therefore acknowledged that what Rabbi Tarfon did was indeed positive, but it was incomplete, just as the Kotzker Rebbe suggested about the child. In the same vein, Ha-Rav Zevin said about Rabbenu Ha-Rav Tzvi Yehudah: "He hasn't done anything". Rabbenu smiled, and Rav Zevin said: "He still has lots and lots to do". And he did! This principle of the Kotzker Rebbe, however, does not have a source. Although the Kotzker Rebbe himself is a source, there is no source for his idea in the Torah, Mishnah, Gemara, Rishonim and Acharonim. It is not written in any place that when a person finishes his task in life, he dies.