Rabbi Herschel Schachter
According to the understanding of Rabbi Nechemia (Zevachim 101a), the chattas (sin-offering) that was burnt (Vayikra 10:16) was the regular korban Mussaf brought every rosh chodesh. The reason Aharon and his two sons were not permitted to eat this korban was because they were all in a state of aninus (the unique form of mourning observed immediately following the death of one of the seven close relatives.) G-d never explicitly forbade eating sacrifices in a state of aninus, but Aharon derived this prohibition on his own through one of the middos of the Torah Shebeal Peh (the oral law): if it is forbidden to eat maaser sheini in a state of animus (Devarim 26:14) because its sanctity requires that it be eaten under conditions of simcha, so much more so (kal vachomer) korbanos, which clearly have a greater degree of kedusha, and therefore a greater degree of simcha, certainly may not be eaten by an onen.
A major premise of the Torah Shebeal Peh is that all the various laws of the Torah must blend in to make one big pattern. The Torah only explicitly states that the kroshim (wooden boards) used for the mishkan had to be placed in an erect position (the same way they grew from the ground), but the oral law assumes that this detail of halacha applies to the lulav and the esrog as well. All religious articles are assumed to have the same halachic status based on this assumption, i.e. that all the laws of the Torah have to fit in to create one big mosaic.
Very often the Torah Shebiksav (the written law) will only provide us with generalities, and the Torah Shebeal Peh will fill in all the missing details (see Medrash Tanchuma to Parshas Noach, no. 3).
According to this view of Rabbi Nechemia, the chumash (here in Parshas Shmini) has recorded the Torah Shebeal Peh drasha of Aharon HaKohen. Moshe Rabbeinu is known for having the highest level of prophecy - direct dictation word for word (unlike all the other prophets), even to the point of "shechinah medaberes mitoch grono". This is representative of Torah Shebiksav. Regarding the development of the Torah Shebeal Peh, just the opposite is the case. The rabbis teach us that "lo bashomayim hi" (see Talmud Temurah 16a), i.e. G-d wants us to develop all the missing details on our own, using, of course, the middos (the principles) of the Torah Shebeal Peh. One of the most important of those middos is the assumption that all of the mitzvos must fit into one pattern.
This principle is generally referred to today as a "derech halimud" (a style of learning). We first try to amass as many details of the halacha as possible, and only then do we set out to try to discern the pattern. We try to understand which halachos - even if they seem to be so diverse - are really connected, and sometimes even identical; and which other halachos, even though they seem to be identical, or at least related, are not really connected at all.
The Talmud (Menachos 29b) records and aggadah that when Moshe Rabbeinu was shown a glimpse of R' Akiva lecturing to his students and developing the many details of the Torah Shebeal Peh, his first reaction was that all of this seemed to be so foreign. After all, Moshe Rabbeinu epitomized and represented the Torah Shebiksav. But ultimately, just as in Parshas Shmini Moshe Rabbeinu gave in to Aharon's drasha of Torah Shebeal Peh, and approved of it, so too in the end he approved of all of the drashas of R' Akiva.
Moshe Rabbeinu represents the earlier period of Torah study (until the days of the Anshei kneses Hagedolah), when the emphasis was placed on the written Torah. From the time of the death of the later prophets (Chagai, Zecharia and Malachi), the emphasis was to be placed on the study of the Torah Shebeal Peh (Seder Olam Zutta). The rabbis in the Midrash Yalkut Shimoni (to Hoshea #625) record a tradition based on a passuk that the geula ho'asida (the future redemption) will be in the merit of our emphasizing the study of mishnayos, i.e. the Torah Shebeal Peh.