What happens when neurons die?

Depends: When the neuron dies there is loss of innervation and that means that what ever it was meant to do it no longer works. The result would be dependent on where and what it controlled.

This does happen with aging. Some individuals do not adapt as well as others and consequently may lose intellectual function and get dementia. A stroke can cause sudden neuronal death, as can toxins. Outcomes are dependent on location and function in general.

In addition to the two very fine responses you have already received I would add this. We used to think that when brain neurons die new ones are not born. We know this is no longer true. But, we need to nurture them by good diet, exercise, lifestyle, so that the new ones flourish and don't die off. See neurogenesis and neuroplasticity searches for more. Peace and good health.