Vayikra: Rishonim:Mitzvah 129 – Concept 362

“But if his means are insufficient for two turtledoves or for two young pigeons, then he shall bring, as his guilt-offering for having sinned, a tenth-ephah of fine flour for a sin-offering;

he shall not place oil on it nor may he put frankincense on it, for it is a sin-offering.” (Leviticus 5:11) We may not place oil on the meal-offerings of wrongdoers. (Rambam, Hilchot Ma’aseh HaKorbanot – The Laws of Sacrificial Procedure)

It is unseemly to adorn an offering that comes to atone for a sin, therefore no oil or incense are placed on it. (Rashi)

The Sefer HaChinuch explains that oil swims to the top of an offering and symbolizes greatness, which is why oil is used to anoint kings and Cohanim. A person seeking atonement should present himself as lowly and contrite.

It may also be that the Torah has compassion for the poor and wants to spare them the expense of oil and incense.