Even though flour offerings normally had oil poured on them, such was not the case for the variable sin offering. The reason is simple: oil was considered prestigious. For example, it was used to anoint High Priests and kings of the Davidic dynasty. But the reason for the sin offering is to humble the person whose negligence required it. To put oil on such an offering would be a mixed message.
An additional reason is that oil was expensive and the flour sin offering was only brought by someone who was so destitute that even pigeons or doves were beyond his means. [The Talmud in Menachos (6a-b) says that, as a sinner, he really shouldn’t get a “discount” on the price of his offering, but it goes back to the first reason, that a sin offering shouldn’t be made fancy.]
This prohibition applies to male kohanim in Temple times. It is discussed in the Talmudic tractate of Menachos on pages 6a-b, 23a, 59a-60b, and elsewhere. It is codified in the Mishneh Torah in the twelfth chapter of Hilchos Maaseh HaKarbanos. It is #102 of the 365 negative mitzvos in the Rambam’s Sefer HaMitzvos.