Well, fundamentally, neurogenesis is just another form of cell division. The most well known disease of cell overproliferation is cancer. Most brain cancers are of the glial cells (i.e. not neurons, but their support cells). There are some exceptions (e.g. some types of medulloblastomas).

A dear family member just started getting symptomatic with a progressive condition where their body generates new motor control neurons (or nerves? I'm not clear), which aren't connected to the brain. It basically amounts to line noise on the motor control circuit, producing incredibly irritating but otherwise harmless uncontrollable motion of specific downstream body parts.

Excessive neurogenesis from supplements almost always lead to things to like glioblastomas, or brain cancer more simply put. You should always be wary of anything related to increased growth or cell division.