Can neurons become cancerous?
Short answer
Neuronal tumors are rare, but they do exist. These cancers develop from neuroblast cells, a population of undifferentiated, dividing precursor cells that will eventually fully differentiate into functional neuronal cells. 

Background
Most neuronal cell types have lost their ability to divide, because of their progressed state of differentiation. There are but a few regions in the brains that generate new nerve cells in adulthood, for example in the hippocampus where neuro-regeneration is believed to be involved in memory formation. Because of the rarity of cell division of neuronal cells in adulthood, neural cancers are rare.

The point where neuronal cells actively divide is during development. These cells are basically still stem cells. A prime example here are retinoblastomas. These are tumors in the photosensitive part of the eye, namely the retina. During early development stem cells actively divide to lay down a layered structure in the back of the eye eventually forming the rods and cones and other visual cells. If this division process goes out of control, tumors can develop. Retinoblasomas can now be diagnosed in utero (Paquette et al., 2012).     

Tumors with a neuronal origin developing post natally are rare, but do exist. Neuroblastoma being a prime example. This is a rare cancer (prevalence of 1:100,000) most commonly found in children younger than age 5. It affects the sympathetic nervous system, which regulates involuntary bodily functions such as heart rate, blood pressure, breathing, and digestion. Neuroblastoma typically begins in the nerve tissues of the adrenal glands but may also begin in the nerves that are located anywhere along the spinal cord, including the neck, chest, or abdomen. The cancer can metastasize (spread) to other organs.

Note that both examples here are called blastomas, those are tumors caused by malignancies in precursor cells, often called blasts, these cells are dividing cells that will develop into a neuron often after a migration phase. Neuroblasts differentiate from neural stem cells and are committed to the neuronal fate. Hence, although these tumors are from a neuronal origin, they are technically not derived from (fully differentiated) neurons.

Reference
- Paquette et al., AJP Rep (2012); 2(1): 55–62