The prohibition against chametz is not limited to the festival of Pesach but has a wider application to the Temple sacrifice: no leaven may be offered on the Temple altar (with one exception, at Shavuot). In next week's sidrah of Tzav, the priests are told they can eat the remains of the grain offering but "not baked leavened".
What is it that disqualifies leaven? Why is it forbidden? One answer that the Rambam gives is that leaven is the medium that other religions use in their sacrifices, and therefore we must separate ourselves from those practices.
Another explanation is that chametz has a wider theological application, by drawing a comparison between leaven and the yetzer hara, the evil inclination, the driving force to sin.
The Talmud records: "Rabbi Alexandri would end his daily prayers with the following supplication: Master of the Universe, You know full well that it is our desire to act according to Your will; but what prevents us from doing so?- the yeast in the dough" (Berachot 17a).
Something becomes chametz when it rises, which is likened to humanity's inflated sense of self, the self-righteousness that drives us to sin.