At first glance, Korach could be written off as just another troublemaker, a minor politician clamoring for an underserved office. And yet, the use of the Divine service - and especially the ketores - as the tool to rid ourselves of him, and the extraordinary miracles associated with his undoing, clearly point to a much greater assault on Yiddishkeit than merely another disgruntled office-seeker.
The Rambam (in Peirush Hamishnayos- hakdama to perek Chelek) indicates that Korach's assault was on the veracity of the Torah transmission of Moshe. He insinuated that the directive to appoint Aharon to be the kohein gadol was Moshe's own design, not Hashem's directive. This therefore made the entire Torah suspect and as such could potentially destroy all of Torah and Yiddishkeit.
Taking a closer look at Chazal, we find that Korach is guilty of another fundamentally flawed perception of the spiritual world of Torah, which is in effect a total eradication of our understanding of Torah and Mitzvos. Chazal (Bamidbar Rabbah 18, 3) tell us that Korach mocked Moshe, asking him, "does a garment that is completely techeiles need tzitzis?", to which Moshe of course replied in the affirmative. Korach then mocked, "if a solitary thread of techeiles is enough to fulfill the mitzvah, isn't an entire garment of techeiles enough?" Korach continued, "does a house filled with sifrei Torah need a mezuzah?", to which Moshe replied, "of course." Once again Korach mocked him, "if two paragraphs of Torah are enough to fulfill the obligation of mezuzah, surely an entire house of seforim is good enough!?"
These arguments are powerful. They fall under the category of "reductio ad absurdum", reducing your opponent's argument to a place of ridicule. What, then, was Moshe's counterpoint? The answer is that Korach's position and argumentation was based on a perception of the spiritual world that was severely flawed. He understood the spiritual world as a finite set of "good" that could be totally obtained by one's efforts. Thus once the "garment" is all techeiles and the house is full of seforim, there is no point in doing anything else, just as a person who works hard exclusively in order to afford a home would find it pointless to work further once he has acquired that home.
But that is a very crass perception of the spiritual world, which sees it as one in which one acquires "things", similar to the physical world. The reality, however, is that the spiritual world is a road leading a person to Hashem. Just as Hashem is infinite, so is the road leading to Him, so to speak. The spiritual is not a specific finite acquisition or set of acquisitions, but a road that continuously leading a person towards Hashem. As soon as a person has progressed down the road, another segment of the road opens up, and then still another segment, and so on ad infinitum. Thus each mitzvah is but a "thread" that leads the person to the beyond; a gateway that opens to another palace, and when one has woven an entire garment of this techeiles, and filled an entire room with seforim, there is a thread that leads him still further, and a door that opens to another room, and so on.
This misconception about the spiritual world is what led Korach to dispute the kehunah in the first place. Chazal say that the reason why it is not common for talmedei chachomim to have children who are talmedei chachomim is in order that people not feel that "Torah" is inherited. For physical entities are always inherited; they are "things" and become the owner's eternally. Not so the world of the spirit. One can have an affinity for ruchniyos but it never becomes anyone's personal possession. Thus Korach's very demand that the kehunah is his disqualified him from that position!