Why did the Two Oldest Sons of Aaron Die?

Rashi: And the Lord spoke to Moses after the death of Aaron’s two sons
What does this teach us [when it specifies ‘after the death of Aaron’s two sons?]’ Rabbi Eleazar ben Azariah illustrated with a parable of a patient, whom a physician came to visit. [The physician] said to him, ‘Do not eat cold foods, and do not lie down in a cold, damp place.’ Then, another [physician] visited him, and advised him, ‘Do not eat cold foods or lie down in a cold, damp place, so that you will not die the way so-and-so died.’ This one warned that patient more effectively than the former. Therefore, Scripture says, ‘after the death of Aaron’s two sons’ [i.e., God effectively said to Aaron, ‘Do not enter the Holy in a prohibited manner, so that you will not die as your sons died’].

Baal HaTurim: The Torah stated at the end of the previous parasha: "My Tabernacle that is amongst them", and juxtaposed to this "after the death." This hints to the Sages’ interpretation on Moses' words to Aaron: "G‑d said, 'I will be sanctified by those nearest Me.'"

Also: The Torah states in the previous parasha: "this is the law" and juxtaposed to this is "after the death" to teach that Nadab and Abihu died as a consequence of issuing a halachic ruling in the presence of their teacher Moses.

Or HaChayim: "and they died"--we focus on the nature of the sons' deaths rather than on the fact of their deaths. The Torah writes, "when they drew near before the Lord" to note that due to their love for G‑d they came too near the celestial source of the light which has a deadly effect on man. This is the mystical aspect of "death by Divine kiss." The deaths of Nadab and Abihu were similar to the death of all other completely righteous men. The only difference was in the case of the deaths of Moses and Aaron, the "kiss of death" of G‑d approaches them, whereas in this case Nadab and Abihu approached the "kiss of death". This is the meaning of the extra letter Vav in "Vayamutu"—"and they died": the Torah alludes to the fact that though these righteous people felt that they were approaching an area which would result in their "kiss of death," they did not flinch and kept getting closer. The desire of their souls to fuse with the divine was so overpowering that they no longer made decisions in which their powers of conscious perception were involved.

The Lubavitcher Rebbe: How can one expect a person at the climax of his spiritual bliss to want to return "back to earth" to his mundane life? If his love of G‑d is genuine, how can he hold himself back at the height of his arousal and re-immerse in the constraints of corporeal existence? From where can a person acquire the vigilance not to go too far?
It depends on how a person starts his spiritual voyage. If he starts from a place of self-satisfaction, he will not want to turn back from his spiritual bliss to attend to the needs of the physical world. But if he wants to follow G‑d's will, even at the high point of arousal, he will be willing to come back down to carry out the mission for which he was created. For G‑d "created the world not to be empty but rather He formed it to be inhabited."
On a smaller scale, every Jew sometimes has a spiritual awakening, perhaps on Shabbat or Yom Tov. At such a time he need to remember whatever he experiences during this special holy moment needs to be taken back with him when he returns to normal everyday life. This spiritual arousal must not be allowed to evaporate without having a tangible effect; it must be harnessed as a moment of true-lasting inspiration. (Likutei Sichot 3:987)

Zohar Acharei 61:
Come and see: the sons of Aaron had no equal in Israel, except for Moses and Aaron. They were called "the nobles of the children of Israel" (Ex. 24:11) and they died because they erred before the Holy King. Did G‑d wish that they should perish? Did we not learn in the secret of the Mishna that G‑d does kindness with everyone, and even evildoers He does not wish to cause perish. So these most saintly ones, would it enter your mind that they should perish from the world? Where were their merits, the merits of their ancestors and also the merit of Moses? How could they have perished?

We have learned from the holy luminary that G‑d concerned Himself with their honor, so inwardly their bodies were tinged with fire, but their soul was not lost, as we have already established. Come and see: even before the deaths of Aaron's sons, it is written, "And Elazar, Aaron's son, took of the daughters of Putiel as a wife" (Ex. 6:25). He was called Pinchas because he was destined to straighten that which is crooked. This is the essence of the verse, "And that which is to be has already been ."