Why don't airliners use noise cancelling?
First, some airliners do use it - the Bombardier Q400 uses a NVS (Noise and Vibration Suppression) system to reduce cabin noise. Basically, it uses devices called Active Tuned Vibration Absorbers (ATVAs) mounted on the fuselage frames to 'cancel' the vibrations from propellers and outside noise, thereby quieting the cabin.

However, there are some issues with using active noise cancellation in aircraft:


The aircraft cabin is quite big and complex, and it would be very difficult to achieve noise cancellation over the entire volume; as active noise cancellation depends on the cancellation of sound waves using opposite phase, it works best in small, confined spaces. If the waves turn out to be in phase in a region, the sound would be doubled - not a good result.
A significant contributor to the noise inside the cabin is the wind outside; this is usually random and the frequency spectrum is also quite wide, resulting in difficulties in noise cancellation.
Noise cancellation is effective mostly when the frequency of the source is constant, which is not usually the case (the props are an exception - they lend themselves well to noise suppression)


Instead of active noise cancellation, aircraft manufacturers are putting more energy in reducing the noise itself:


Better engines, which has lesser noise through the use of chevrons, etc.
Better modelling of aircraft, so that the fuselage design minimizes the noise due to wind.
Better damping and cabin design, so the noise produced (for example, due to air conditioning and ventilation) is reduced.