Absolutely, positively no.

The average Italian citizen in 1310 would have been a barely literate farmer. He might understand some references if they referred to individuals they had interacted with; for example, it is likely that a peasant in Romagna would have heard of Malatesta da Verrucchio, namely because he murdered and plotted his way to the Lordship of Rimini, and consequentially, probably would have known that Malatesta's son Paolo son was killed by a cuckolded husband.

But statistically speaking, most references would just be too specific. If a similar book were written about your country, would you understand a similarly staggering amount of references to local politics?

For example, the people identified in the Sixth circle and the subsequent discussion on the infighting in Florence would probably be comprehensible to your average working class Florentine, but as for the "Average Italian"; unless they were working the land of one of the families involved/mentioned, they probably would have no idea what the references were about.

The references about classical literature and history would also be completely lost on anyone who did not have a classical education.

Of course, Italy did have a larger bourgeoisie than most European countries at the time, but it was still a relatively small part of the population, and the difference between even these individuals and the poor would be staggering today. Further, it is is conceivable that a burgher, educated in a local parochial school, would understand the classical references, it would be still unlikely he would have more than a basic understanding of the politics referenced unless he was a Tuscan.

Dante wasn't writing down popular tales or fireside stories. He was writing a poignant condemnation of contemporary politics. Dante Alighieri was writing for a hyper-educated elite which had cast him out, disregarding his loyalty to his city (incidentally, this is also why his references go so far beyond tuscan politics. The courts at Ravenna, Rimini and Verona were more than happy to welcome him). The Divine comedy, as a study of Morality, damns Dante's enemies, using a complex system of allegories to show them just how immoral they are, and touts the virtues of those who had welcomed him. These were highly elaborate political games that pitted family against family, all fighting for dynastic supremacy in the mobile world of the early italian renaissance. The Divine comedy reflects this.