Why did Aaron get praised when he was silent upon hearing of the death of his children - wouldn't that seem to be very cold and unfeeling?
R' Mordechai Greenberg (Rosh Yeshiva of KBY) once explained that the idea of a Kohen is to be the shaliach (proxy) of the nation in their relationship to Hashem. They bring the offerings of the people, and are meant to be faithful representatives of those whom they represent. Thus, everything that the Kohanim did was emphasized as being "כאשר צוה ה" - exactly as Hashem commanded. A Kohen who is putting his own feelings and emotions into the Avoda is warping his sender's own service. Therefore, Nadav and Avihu had perilously endangered the entire institution of Priesthood and its functionality by exuberantly bringing a fire "אשר לא צוה ה", that Hashem had not commanded.
Aharon was actually saving the entire institution of Priesthood with his silence - he was showing that one is capable of being unmoved by his own emotions and dutifully doing exactly what is expected of him.
Silence is Golden by Rabbi Shmuel Silber According to some he was so shocked that he was emotionally paralyzed. He couldn't react, he couldn't even cry. According to others his faith was so solid and unshakeable he forced himself to continue with his duties. We see from his burning the Rosh Chodesh Musaf that he did indeed react.
Aharon’s reaction to this is silent acceptance of the realities that now face him. He does not rail against perceived injustice, as does Iyov. Nor does he withdraw from the fray of life and go into seclusion, as did many others when faced with similar tragic situations. Aharon becomes the paradigm for how humans are to deal with the “eighth day” – with life and its ups and downs.
Resilience and silent inner strength engendered by faith and acceptance of God’s will are the weapons of living on in spite of all that the “eighth day” imposes upon one’s life. These words are much easier to write and to read than to actually implement. Yet the Torah expects no less from us than it did from Aharon. Life and our contributions and meaningful behavior towards making it better and stronger are always played out on the background of the “eighth day.”
As for Aharon's silence, it is most fitting of all. Who realized more than the aged priest the great dangers inherent in unbridled religious enthusiasm? Who understood more than he did, as he stood powerless to stem the crowd excitedly clamoring for a golden calf, that the best intentioned acts of worship can so easily degrade into coarse and corrosive idolatry? His beloved children, whom he hoped would one day succeed him in the service of God, now lay dead at his feet, but how could he blame the Deity for their demise? Nadav and Avihu were consumed by an inner fire that could not but be matched by an outer fire from above. Did the poet king perhaps have them in mind when he bid his beloved to
Place me as a signet upon your heart and as a seal upon your hand, for love is as powerful as death and yearning as cruel as the grave, its flames are the flames of God's consuming fire! (Shir ha-Shirim 8:6).
And did God Himself perhaps have them in mind as well when He twice warned Moshe at the revelation at Sinai to distance the people and the priests from the mount, lest "they break forth to ascend towards God, and He then break forth upon them" (Shemot 19:24)?
All religions speak of Divine mercy and love, inviting us to lovingly embrace God in turn. And who is there that does not want to experience that connectedness and concern of God in their own life? Who is there that does not want to serve God sincerely with enthusiasm and passion? One of the most unique contributions of the Torah to religious discourse, however, is that worship of God, if it is to be genuine, must also be predicated upon reverence and fear, for reverence provides the restraints that love refuses to recognize. Nadav and Avihu tragically perished at the very moment that God's presence was most intensely felt. But through their death, the two brothers communicated a fundamental lesson that was to inform the service of the Mishkan and Temple (and for that matter the synagogue) for all time. "Moshe said to Aharon: this is in accordance with what God spoke, for He said that 'I will be sanctified by those that are close to Me, and I will be glorified in the sight of all of the people'."
Now we can see why the Torah must comment on the fact that Aharon is silent. Aharon has learned the lessons which Hashem had just taught and accepted Hashem's will. The Torah develops this idea of maintaining a distance from Hashem in the rest of Parshat Shemini by contrasting purity and impurity in the later Pesukim. This is also why Hashem says "And separate between Kodesh and Chol, and between impure and pure."
Ramban: Aharon had been weeping, but on hearing Moshe's consolation (Shmini 10:3) he stopped.
Seforno: He found comfort in the knowledge that Hashem's Name had been sanctified by his sons.
Rashi: As a reward for his silent acceptance of Hashem's decree, Aharon was honored by having the following mitzvos (10:8-11) addressed to him.
Rav Shimshon Refael hirsch points out that this is an indication that the greater the people, the more strict Hashem is with them. As a result, Aharon could not respond because he realized that Hashem must answer the slightest deviation of his great ones. Rav Hirsch also points out that this is one of the many things told to Moshe that were not written down and, if not for this incident, would have remained in Torah Sheb'al Peh.