Wiki confirms no oil in sweat. Seems reasonably accurate.
"Perspiration (also called sweating or sometimes transpiration) is the production of a fluid, consisting primarily of water as well as various dissolved solids (chiefly chlorides), that is excreted by the sweat glands in the skin of mammals."

Regular cleansing of face and body is a good thing.

When I was in high school back in the Dark Ages, a senior (I was a sophomore) shaved his head and was expelled. He probably would have been executed for having a Mohawk.