For the vast majority of us, burning fat whilst at the same time building muscle is possible to achieve. How much will greatly depend upon your genetic makeup - whether you are more prone to fat loss and/or muscle gain.
Generally speaking, the more fat loss you experience, the more of a calorie deficit your body will endure. This will discourage protein synthesis. So at the end of the day, you need to find a happy medium between fat loss and gaining muscle.
Speaking from personal experience, about a year ago I entered in a 12 week competition where I lost 9.9kg of fat mass and gained 0.8kg of lean mass (as detailed here). This was achieved by a sound eating plan, alongside consistent high intensity training.