Nadav and Avihu died before Hashem when they offered an alien fire before Hashem, and they had no children (Bamidbar 3:4). Four times the Torah mentions the death of Nadav and Avihu, as well as their sin. This indicates that this was their only sin, so that people should not say that they committed terrible sins secretly for which they were punished (Yalkut Shimoni 624).
Why was the Torah afraid that people would suspect Nadav and Avihu of "ma'asim mekulkalim baseiser," of hidden destructive acts? Apparently, prominent individuals sometimes consider themselves above the law. Nadav and Avihu were considered greater than Moshe and Aharon (Rashi Vayikra 10:3). Their untimely death, despite their outward greatness, aroused suspicion of clandestine indiscretion (see Shabbos 13b). The Torah testifies that their only sin was the offering of an alien fire.
Remarkably, the Yalkut Shimoni proceeds to enumerate an entire series of causes that led to Nadav and Avihu's death. While this may represent a Midrashic dispute, the Yalkut may be suggesting a series of character flaws which led to the single sin and the ultimate tragedy.
Let us begin with the causes based on the last phrase of the aforementioned pasuk in Bamidbar - and they had no children. One who is blessed with children has a greater sense of responsibility and is less inclined to make reckless mistakes. Perhaps if Nadav and Avihu had children, they would not have taken chances. The risk of leaving behind orphans might have prevented their sin.
R' Levi said that Nadav and Avihu never married. Many women were waiting for them, but they said "our uncle (Moshe) is the king, our uncle (Nachshon) is a prince, our father (Aharon) is the kohein gadol, we are his assistants, what woman is worthy of us?"
While we must approach biblical giants with appropriate deference, the Yalkut is clearly teaching us lessons for all generations. Greatness, whether inherited or achieved, can breed arrogance. This negative trait, which prevented Nadav and Avihu from getting married and having children, convinced them that they were above the law forbidding alien fires. Moreover, a wife, to whom they would be accountable, could have saved them from their fatal decision (see Sanhedrin 109b).
Failure to consult contributed to the death of Nadav and Avihu as well. The Yalkut teaches that they did not consult Moshe Rabbeinu or respect their father Aharon. In addition, they issued halachic rulings in the presence of their rebbe, a sin punishable by death.
Moreover, their greatness led them to grossly inappropriate impatience. "Moshe and Aharon were walking ahead. Nadav and Avihu walked behind them and said, 'when will those two elders die so that we can rule in their place.'"
While another view holds that they did not verbalize these thoughts, their attitude led them to their fatal error. Nadav and Avihu, despite, or perhaps because of, their greatness, did not know their place - in the mishkan or in the hierarchy of leadership.
Finally, and most incredibly, Nadav and Avihu did not consult one another. Each entered the innermost sanctum (kodesh hakodoshim, Zayis Ra'anan on the Yalkut, no. 17) independently. The Midrash implies that had they consulted one another they would not have sinned.
Even consulting a peer provides a measure of humility which could have prevented their misdeed. A discussion of the plan might have revealed the dangers inherent in it, which were overlooked in the individual and private musings of Nadav and Avihu. In addition, each brother may have brought the alien fire in an attempt to be the greatest kohein. The knowledge that they would have to share this status would likely have averted the disaster altogether.
Notwithstanding the character flaws detailed in the Yalkut Shimoni, the basic desire to offer the alien fire stemmed from an unquenchable thirst for greater ahavas Hashem. This led Nadav and Avihu to cross the boundaries of halacha and bring an unauthorized ketores. A similarly fatal mistake was made by the two hundred and fifty men who offered the ketores when Korach rebelled. Their desire for closeness to Hashem was so strong that they were willing to give up their lives (Netziv Vayikra 9:6, Bamidbar 16:1).
Nadav and Avihu were concerned with their own spiritual advancement. As they tried to raise themselves above the klal, Hashem's name was sanctified through them as they were separated from the klal (Meshech Chochmo Vayikra 10:3).
As we strive for spiritual growth, we should do so in order to better serve Klal Yisrael, by teaching and by example. This attitude instills the requisite humility which prevents the violation of halachic boundaries. Indeed, no man is above the law.