Will neurons die if they are inactive for a long time?
Short answer
Yes. 

Background
Postsynaptic neuronal degeneration after deafferentation are notorious in the peripheral sensory systems. There, marked degenerative processes are present when input to the secondary neurons is lost due to degeneration of the primary sensory neurons. In other words


When the hair cells are lost in the cochlea, the auditory nerve degenerates (e.g., Kuyama, 2010);
When the retinal photoreceptors are lost, for example due to degenerative diseases like retinitis pigmentosa, bipolar and horizontal cells in the retina degenerate due to a loss of efferent input (Mazzoni et al., 2008). For more information see the works of Marc &amp; Jones, e.g.., Jones &amp; Marc (2005).  


Note that in the central nervous system these effects play less of a role, as neurons have ample opportunity to reconnect to other neurons, for the better or worse; e.g., in the hearing impaired and visually impaired, spontaneous sensations of sound and visuals may develop and lead to tinnitus and photopsias, respectively. These symptoms are caused by cortical sensory neurons desperately and randomly looking for sensory input after being deafferented due to peripheral loss of hearing or vision. When they cannot obtain useful input any longer, they gather irrelevant, non-useful input and start generating spontaneous 'illusionary' perceptions instead.   

References
- Jones &amp; Marc, Exp Eye Res (2005); 81(2): 123-37
- Kuyama, J Neurosci (2009); 29(45): 14077–085)
- Mazzoni et al., J Neurosci (2008); 28(52): 14282–92