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.
Q: Does extra virgin coconut oil require hashgacha?
A: Extra virgin coconut oil does require 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.
It may be due to equipment issues and cross-contamination with non-kosher productions.
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.
Per Rabbi Yitzchak Abadi the coconut oil is fine without a hecsher. (Same with sesame seed oil, which is not just his opinion but the opinion of the Rema.)
If you were to ask Rabbi Abadi why kashrut agencies would tell you it needs a hecsher for fear that it might get contaminated with other products, I assume he would tell you that its part of the ways in which hechaher companies try to make money by keeping people ignorant of the halacha. He would then encourage you to read the shulchan arukh to find out what the halacha is, rather than listen to the guidelines designed by the hecsher companies that are far beyond what the halacha requires.
Source for the coconut oil: http://kashrut.org/forum/viewpost.asp?mid=53596&highlight=coconut oil