The idea is to eat fats that are good for the body and to eliminate fats that are harmful to the body.
Rather than melting coconut oil into tea, use it as you would otherwise use lard, margerine, shortening, butter, corn oil or other fats. A bit on broccoli, a spoonful as you cook your whole grains, etc. It is terrific on popcorn and some people spread it on their toast in the morning. Saute your onions in coconut oil. It is easy to use a few tablespoons in normal meal preparation.
The body needs about 30% of its calories from good fats. Coconut oil is one of the good fats. It doesn't supply omega-3 fatty acids, so you need another source for that.
The body needs no transfats or animal fats. Deep fried and processed foods have loads of transfatty acids which are poison.
Eliminate the bad fats, eat the good fats and your thyroid condition will improve, your weight will drop, and you will feel better.