After getting the acknowledgement of your Italian citizenship, for any reason, you do not want to live in Italy.
You are able to live in anywhere, since you warn the Italian consulate in the country you intend to live. That’s possible because before entering into the process, you had to proceed with your residence permit in Italy. For that reason, there is AIRE - Anagrafe deegli Italiani Residenti all’Estero, or Civil Registry Office for Italian Citizens Living Abroad.
The “Anagrafe,” if you are used to the documents that compose the process to acknowledge the citizenship, it is a document certifying your residence in Italy. In other words, AIRE is just a notice to Italy that such citizen does not live in the country.
It is free of cost and mandatory if you are going to effectively live in another country for more than 12 months. It is used also if an Italian newborn was conceived abroad. It is important to remember that keeping the AIRE up to date is a duty of an Italian citizen. Information as address, civil status, notice of return to Italy or loss of citizenship are some of the main reasons.
And this process is very simple! You only need to ask for an AIRE request in your region, fill it out and return it. By that, the consulate will be aware of that citizen lives abroad.