An exception to the above rule is extra virgin olive oil, which is processed without heat and is acceptable without hashgachah. There are persistent reports of extra virgin olive oil being adulterated, and a more careful reading of the data shows that the vast majority of the suspected adulteration is in ways that might affect the quality and uniqueness of extra virgin olive oil, but not in a way that poses a kashrus concern. Other forms of olive oil (including extra light) require hashgachah just like other cooking oils.
Oils and fats consist of two parts: one “glycerin” molecule connected to three “fatty acids”. [The term “oil” is usually used when the item is liquid at room temperature, and “fat” refers to those which are solid at room temperature]. When subjected to very high temperatures, these two parts can be separated, after which the individual parts are each purified (again, at high temperatures) and used in many products. Glycerin derived from animal fat is chemically identical to that derived from vegetable sources (and the animal-based glycerin tends to also be cheaper), and even vegetable-based glycerin might have been produced on equipment also used for animal products. Thus, items which contain glycerin are particularly kosher-sensitive. Fatty acids are likewise kosher-sensitive, and they are used as raw materials in the flavor industry and elsewhere.
Oil is not only used for cooking, but is also a lubricant, and when that lubricant has food contact it can only be used if it bears reliable hashgachah. Some examples of this are Pam (cooking spray), and parchment paper (Quilon). Silicone coated baking paper does not require certification because the silicone is innocuous from a kashrus perspective. Similarly, mineral oil does not require certification (it is derived from petroleum) and lubrication made of that material is inherently kosher.
Even if animal fat/oil is derived from a kosher animal we assume that it is non-kosher, because the animal did not undergo shechitah. Fish do not require shechitah and therefore this concern does not apply to them. Accordingly, oil that comes from a kosher fish is kosher, but one must be certain that the oil actually comes from a kosher fish, and that it was not processed on the same equipment as non-kosher oil from fish or animals. Due to these concerns, fish oil requires kosher certification.
Essential oils, such as bergamot oil or eucalyptus oil, are derived from botanicals and tend to be free of kashrus concerns. There are some exceptions, such as grapeseed oil which requires hashgachah, and one should check with the cRc office before using a given essential oil. [Oleoresins, absolutes, and other botanical extracts do pose kashrus concerns, and should only be used with hashgachah.] Similarly, citrus oils, such as orange oil or lemon oil, are typically acceptable without hashgachah, but it is reasonably common for them to come from Eretz Yisroel which means that one must be alert for concerns of arlah, shemittah, terumah, and ma’aser.