I am specifically referring to Coconut Oil. I know that my question is similar to this one: Is Extra virgin olive oil kosher by default? but My question is also about non-olive oils. Due to the huge olive oil industry there is a large profit to be made in manufacturing fraudulent products which is part of the analysis as to whether or not to permit (extra virgin) olive oil without a hechsher. In addition the product I have says: Free of: Milk, eggs, peanuts, shellfish, fish, soy, gluten, titanium dioxide. The specific product I am referring to can be seen here: http://www.vitacost.com/vitacost-extra-virgin-certified-organic-coconut-oil-54-fl-oz-10
Per the CRC-Chicago it requires a Hechsher.
Rabbi Abe Sharp responded to my e-mail sent to the CRC-Chicago why a Hashgacha is necessary for extra virgin coconut oil.
Based on the OK article I mentioned in this answer, coconut oil would need a hechsher.
According to the Wikipedia article on coconut oil, all the different methods of creating coconut oil involve cooking or baking the coconut, and some of the methods involve additives.
The OK article discusses the concerns involved in oil production, and how Extra Virgin Olive Oil is the exception, since it is a cold press with no further processing.