Moshe emphasizes that the legitimacy of his position would only be established if Hashem causes the ground to open its mouth and swallow them up and bury them alive (Bamidbar 16:30). Why was it specifically necessary that Korach and his followers be buried alive?
The Ponim Yofos explains based on a Gemara in Shabbos (105b) which teaches: “somebody who fails to properly eulogize a talmid chocham deserves to be buried alive”. The Medrash Tanchuma answers that although the Jewish people willingly accepted the Written Torah, they were hesitant to accept the Oral Torah. Therefore, Hashem lifted the mountain over their heads to coerce them to additionally accept the Oral Law.
The underlying challenge posed by Korach was a refusal to accept the validity of Moshe’s traditions that he heard directly from Hashem, namely that he should serve as the leader of the Jewish people, and Aharon should function as the kohen gadol. This represented the “Oral Law” of Moshe, and by disputing it, Korach and his followers deserved the same punishment given to others who reject the Oral Torah: They were buried alive.