Well, even if fat consumes more energy to utilize as fuel than carbs or protein, that doesn't mean fat is an energy free food. It is nonsensical to say eating more total calories, regardless of where they come from, will speed up weight loss. Maybe if you were eating so few calories before that your body was starving and slowing down all body process, I could see eating more calories in this case might encourage your body to burn fat. But if you are eating enough already (and unless you FEEL very hungry, you should be), eating more calories will only decrease the amount of body fat your body needs to burn.