I always liked former director of the United States Holocaust Memorial Museum Walter Reich's take:

The primary motivation for most deniers is anti-Semitism, and for them the Holocaust is an infuriatingly inconvenient fact of history. After all, the Holocaust has generally been recognized as one of the most terrible crimes that ever took place, and surely the very emblem of evil in the modern age. If that crime was a direct result of anti-Semitism taken to its logical end, then anti-Semitism itself, even when expressed in private conversation, is inevitably discredited among most people. What better way to rehabilitate anti-Semitism, make anti-Semitic arguments seem once again respectable in civilized discourse and even make it acceptable for governments to pursue anti-Semitic policies than by convincing the world that the great crime for which anti-Semitism was blamed simply never happened— indeed, that it was nothing more than a frame-up invented by the Jews, and propagated by them through their control of the media? What better way, in short, to make the world safe again for anti-Semitism than by denying the Holocaust?

Source: http://www.nytimes.com/1993/07/11/books/erasing-the-holocaust.html

The reasons I've seen are as follows:

• Anti-semitism (the most obvious one)
• A deep-seated, irrational fear of authority to the point where authority is assumed to be lying about everything even remotely important (/r/conspiracy)
• A need to feel like there is an "other" that is victimizing "us," which to some people begets a need to assert that the "other" is purely evil and thus must not have been victimized (ties into the antisemitism, of course)

To add to the existing comments, the work on the sociology and psychology of those people who support right-wing authoritarian ideologies by people like Robert Altemeyer and Karen Stenner suggests that such people often engage in a variety of mental contortions / gymnastics in order to wedge their worldview and political preferences into the real world.

In the case of Holocaust deniers this can take the form, roughly speaking, of the idea that a strong leader at the helm of a "soft-totalitarian" police state doesn't drive the sort of degeneration of civil rights which leads to the mass opression and subsequent incarceration and murder of the citizenry. The "logic" being based on the "wild exaggeration" of the various crimes against humanity perpetrated by the Nazis during World War II; therefore the most common objection to these forms of governance is invalid because it's based on "false" premises.