When Dante was convicted in 1302, Florence was the Republic of Florence. The Republic of Florence was subject to the Holy Roman Empire, but still remained a distinct legal jurisdiction (like a state.) The Republic of Florence was replaced by the Grand Duchy of Tuscany in 1569, which ruled Florence (with some interruptions) until 1859, when it was conquered by the Kingdom of Sardinia, which soon became the modern Kingdom of Italy, which in turn became the present-day Italian Republic after WW2. Each successor entity inherited the legal rights of its predecessor entity – the Italian Republic inherited the legal rights of the Kingdom of Italy, which in turn inherited the legal rights of the Kingdom of Sardinia, which in turn inherited the legal rights of the Grand Duchy of Tuscany, which in turn inherited the legal rights of the Republic of Florence. One of the legal rights of the Republic of Florence was to rescind the legal judgements of its courts, even posthumously. Hence, that legal right has been inherited, through a chain of successors, by the modern-day Italian Republic, which therefore has a unique legal authority (not possessed by any other jurisdiction) to formally rescind Dante's conviction and sentence. Yes, it is a right of no practical effect, its exercise would be of purely symbolic consequence, but it would be the unique possession of the Italian Republic.
(The succession story I presented – Republic of Florence => Grand Duchy of Tuscany => Kingdom of Sardinia => Kingdom of Italy => Italian Republic – is no doubt somewhat simplified, but I don't think fleshing out those added details would change the ultimate conclusion.)