Coconut Oil can be taken on an empty stomach if you want too but it's best to take the coconut oil after you've eaten your breakfast for the full health benefits.
You can also add some Coconut Oil to some Tea or Coffee and still reap the health benefits of the coconut oil.
If you do take some Coconut Oil on an empty stomach it might cause diarrhea which also makes Coconut Oil a good remedy for constipation.
If you have constipation you should take the coconut oil on an empty stomach to help get your bowels moving but for the other health benefits of coconut oil it's best to take it after you've eaten breakfast or any other meal.