1) There are some dividing neuroblasts (cells that make more neurons) in at least some adult vertebrates. These have been shown to be able to replace some adult neurons, like some olfactory (smell) system neurons and some hippocampal neurons (involved in learning and location). These maybe the only ones.
If new stem cell technology can make different kinds of neuroblasts, more kinds of neurons might be replaceable.

2) If you made one's neurons immortal (but constrained to be metabolically supported in a mortal body) as Berkeman said. Ultimately the body would breakdown and the brain would die. Then it might be possible to support the body with some fancy equipment, but even then, the brain made of many non-neuronal cells, will eventually have problems (like a cancer). Nothing is going to last for ever.

Evolution's answer to the limits on individual persistence is to make copies by reproducing. This makes copies of the maternal system that are very similar functionally to their mother. Close functional replacement is about as good as you can do.