That which remains of it shall be eaten by Aaron and his descendants… (Leviticus 6:9)
Only a small handful (three-fingers’ full) of a flour offering was burnt; the rest was to eaten by the kohanim. An exception to this rule is that the flour offering of a male kohein was burned in its entirety and not eaten (as per verse 6:16). However, a flour offering brought by a kohenes (a woman of a priestly family) is dealt with in the normal fashion.
As we mentioned in Mitzvah #102, the underlying rationale for this mitzvah is that it raises the prestige of these sacrifices that they may only be eaten by the kohanim, who minister in the Temple.
This mitzvah applies to male kohanim when the Temple service is in effect. In the Talmud, it is discussed in the six chapter of tractate Menachos, starting on page 72b. It is codified in the Mishneh Torah in the tenth chapter of Hilchos Maaseh HaKarbanos and is #88 of the 248 positive mitzvos in the Rambam’s Sefer HaMitzvos.