Does learning increase the number of neurons in the brain?
Short answer
Training increases white and gray matter densities in the brain. This may reflect increases in neuronal cell counts especially in the hippocampus. In the cortex, however, such observed changes are probably more reflective of other processes, such as synaptogenesis.   

Background
Gray matter is generally viewed as being the neuronal cell bodies, while white matter corresponds to myelinated fiber tracts. Changes in both can be reflective of changes in cellular content, but changes in gray matter are generally more directly related to changes in cell numbers, which you are after.

A striking example of the effects of learning on the brain is observed in subjects that start to learn to juggle. These people have been shown to have increased gray matter density in the occipito-parietal regions (the visual motion areas). These changes are apparent after as little as 7 days of training. Such experience-dependent changes are also observed in white matter. Occipito-parietal regions are involved in visuo-motor coordination, reaching and grasping. There have been a host of other studies that showed similar changes in the brain. For example, taxi drivers were shown to have increased hippocampal volumes. 

There is good evidence for adult neurogenesis occurring with learning in the hippocampus. Adult neuro­genesis also produces thousands of new granule cells in the dentate gyrus every month, but this is a relatively small increase in total number of hippocampal neurons. Furthermore, although there have been reports of neurogenesis in the mammalian adult neocortex, this is controversial.
Thus, neurogenesis is likely a minor factor in MRI changes, particularly those found outside the hippocampus.

Instead, other factors such as increase in glial cell numbers, synaptogenesis, changes in dendritic structures, axonal sprouting and re-routing and even vascular changes may underly the observed changes in imaging studies.    

Note: there is a host of imaging studies available on effects of learning on structural changes in the brain. The referenced review article cites many. 

Reference
- Zatorre et al., Nature Neurosci (2012); 15: 528–36