Renouncing US citizenship is always a sensitive subject, followed by lots of myths and misconceptions.
There are certainly many advantages to it, but also some downsides, depending on individual cases.
The biggest and most obvious benefits of renouncing citizenship of the United States are tax-related. Also, you would have more freedom to go and do business offshore, but you would still be able to bank in the US, even if you're not their citizen anymore.
On the other hand, you will no longer have access to the US job market and you'll most probably need a visa to travel to the States, so think about your options carefully and be prepared for this change.