Will neurons live after the death of a human? If so, how long?
I think that depends somewhat on how the human dies - in particular on the conditions. If they die in a very cold environment (say at the top of some icy mountain range), they'll freeze and metabolic processes will slow down considerably, and so will cell death. But such hypothermia-based slowing down is a special case. Normally, if your heart stops or you are not getting oxygen to your brain (perhaps due to a stroke), your neurons will start dying pretty quickly. Within many minutes there will certainly be considerable permanent cell loss - certainly within hours. If oxygen is significantly depleted, ATP production will also significantly drop. If this happens, the various ATP-dependent processes will slow to a crawl (including the Na/K ATPase and other ion pumps and channels that are needed for maintaining and changing the voltage difference between the inside and outside of the cell). This in itself is bad enough as a lot of the enzymes and mechanisms that are dependent on the voltage difference will fail (and probably enough of a reason for many cells to start dying), but one of the effects of this is apparently an increase in the extracellular glutamate, which ultimately leads to (via NMDA receptors for example) an influx of calcium ions into the neurons. Too much of the calcium ions is once again toxic for a variety of the processes inside the cell and is an additional and big reason for neuronal cell death.

In short, as far as I know, neurons don't have a good time living without sufficient oxygen and therefore after a typical human death.