Pure fat won't affect insulin. Coconut oil can be used to reduce the stress (if any) since it quickly feeds the brain and reduces glucose demand. I think Paul Jaminet mixes it with spinach and lemon juice to have something to munch on without interrupting the autophagic benefits.
However, all energy goes somewhere so it will cost you some fat loss--the amount matters.
Lots of: urban hiking, cycling, sprinting
Lots of: fresh meat, seafood, eggs, organs, tubers, starch fruits, vegetables, meat fat, dairy fat, oil fruits
Some: cured meat, dairy protein, sweet fruits, rice, pulses, tree nuts, oil seeds
Minimal: soy, refined proteins, sugar, liquid carbohydrate, grains, refined oils, peanuts