First off, remember that everything "burns" on a continuum. You don't switch from one source to another like flipping a lightswitch on & off.
Next, you should realize that breaking down protein & using it as fuel is not popular with your body. Unless you're starving (literally), or you're doing ultra long duration aerobic exercise (think 4+ hours of running or riding at an effort that's not incredibly easy), only about 8% of what you break down for fuel will be protein (it can creep up to 15-20% in the aforementioned situations).

There is some truth to those charts at the gym that tell you to exercise at a lower intensity to burn fat, because it takes more available oxygen to break down and mobilize fat. Working at lower intensity levels fits this bill. Also, at very high intensities, a low (acidic) Ph in the blood can prevent fat from getting mobilized from the adipose tissue to the muscle in need.

However, don't forget the continuum- when you're exercising at a moderate to high intensity, your energy demands are much greater, so you're likely to burn more of both fat and carbohydrate for fuel, even if you are going a little heavier on the carb side than you were at low intensities.
And yes, you do need some carbohydrates to burn the fat- this just means that during extended durations (50+ min) of moderate to high intensity exercise, you should supplement with some sort of carbohydrate- drink, gel, food, whatever works for you. At those intensities, there is a risk of getting low on carbohydrates if you don't keep them coming in.