If there were to be such a museum in Italy, it would have to be in Florence, where Dante almost single handedly made the medieval Florentine dialect into the national language of Italy (in writing the Divine Comedy, he had been tempted by Occitan, the language of the troubadours, before making his fateful decision).
Due to open in 2023, it promises rare editions, audio vistutal and exhibits on the development of Italian from Latin. Check the website on the museum’s current status,
Piazza della Stazione 6
Image by Internet Archive Book Images