The answer is it depends. If you renounce U.S. citizenship, and then get a residence permit, you will be able to live in the U.S. because you have the residence.If you do not have the residence permit, you will be staying illegally, ex-citizen, or not.
Now it also depends on the time frame. If you plan on coming to the U.S. and staying on temporary visa and doing a visa hop every so often, some people are able to temporary live in the U.S. with this method.
But, if you plan on renouncing U.S. citizenship, don’t expect to still be able to live in the U.S., unless you get sometime of residence or visa on your new documents.
I hope this helped. If you want to learn how to renounce US citizenship, but you only have 1 citizenship, there is a way to renounce U.S. citizenship and exchange it for Permanent Residence in an EU country. Visit Residencyelite.com for more information.