There's a lot of problems that can come about as a function of celebrity. It's almost a trope that child stars end up abusing drugs and spiraling into depression, so far removed from the realities of reality that they have no context for when people are taking advantage of them or they find themselves in abnormal situations, or just have no sense of moderation or self-awareness as to what may be good for them or bad for them.
Narcissism is another one. Cue Kanye West's current foray into starting his own religion, or maybe his own service to an existing religion. It's too obscure for me to tell right now which is which or what his overall intentions are, but if Scientology is any kind of template, then the religious movements that spring out of celebrity aren't great. Not that the orthodox ones that come about by more traditional means are any better.
But perhaps the most glaring and unavoidable problem with celebrity is that if you become famous, especially as a performer, performance is now your life. You perform for money. Your skills, or talents, or presence, snippets of what you say online or in life, actions that you take in the spotlight or out of it, can all be construed as a part of your performance. Accurate or not, for better and for worse. I imagine it can give one a deep sense of loss of personal agency, which may actually be true in a lot of cases.