The kohanim were to eat the remainder of the flour offerings but they were limited in how they could prepare it. They were not permitted to bake it into leavened bread. The Mishna in Menachos tells us (page 55a) that the flour was kneaded with warm water, with the result that it had to be watched so that it didn’t become chometz – just like when making Passover matzos!
The Sefer HaChinuch does not discuss the underlying reason for this mitzvah but we have previously pointed out the negative character traits symbolized by chometz. Foremost among these is ego. For such reasons, leaven could not be brought on the altar (see Mitzvah #117). Presumably, such symbolic eschewing of negative characteristics is likewise the reason for the current prohibition.
This prohibition applied to everyone, both men and women, while the Temple stood; even though only kohanim ate the flour offerings, no one was permitted to leaven them. This mitzvah is discussed by the Talmud in tractate Menachos on pages 55a-57b and is codified in the Mishneh Torah in the twelfth chapter of Hilchos Hilchos Maaseh HaKarbanos. It is #124 of the 365 negative mitzvos in the Rambam’s Sefer HaMitzvos.