Is it safe to wash items that have been exposed to gasoline or cooking oils?

No — the manual includes a fire hazard warning stating you should never place items in the washer that are dampened with gasoline or other flammable fluids. No washer can completely remove oil, so you should not wash or dry anything that has ever had any type of oil on it, including cooking oils, as doing so can result in death, explosion, or fire. (Page 1)