Training first thing in the morning doesn't make you burn any more fat than you would any other time of the day.
the theory for why you would burn more fat at that hour is because you already have low blood sugar from having not eaten in 8+ hours, so to keep you conscious your body has to use other sources of energy, with fat being the next most available form.
Except in practice, if you build up any appreciable intensity, your body needs an anaerobic energy source, so it'll just eat up whatever blood sugar you have remaining, and you're prone to falling unconscious. There's a true story here. Likewise, being starved, you won't be able to give the best performance, anyway, so there's less benefit -- both in calorie usage and fitness enhancement -- than training at another time.
Furthermore, in the context of a 24 hour day, and a 168 hour week, the half an hour or so that you might be able to train first thing in the morning before needing to eat/pass out is relatively insignificant as far as ratios of fat/carb burning go. It doesn't normally matter whether the energy you're burning within a session is from carbs or from fat. 24/7 you're building up and breaking down your fat cells, just as how 24/7 you're building up and breaking down your muscle cells. Most of that time, the majority of energy is coming from fat. In the times where increased carb usage occurs, that just means that those carbs won't be turned into fat later on, so the building up of fat cells is reduced, and if the breaking down of fat cells remains constant, then you've shifted the balance in favour of fat loss.