TLDR: Shakespeare was clearly familiar with a lot of Italian literature second-hand, and there is circumstantial evidence for first-hand.
Shakespeare's Italian influence is a question that's aroused a lot of interest. How could it not, when a third of his known plays were set at least partially in Italy in a wide range of geographical locations? Indeed, it has been noted that he correctly included a lot of local details in his scenes, such as church names and routes via canal. Yet this is in an age when international travel was still relatively uncommon. There is no evidence that Shakespeare ever went there, and his relatively modest wealth compared with the aristocracy would have made it very unlikely.
We do know that Shakespeare was intimate with at least one person who had travelled to Italy: the actor Will Kemp who was part of the same company as Shakespeare. He is documented to have travelled in both Germany and Italy sometime after 1599.
Italy was inspirational among poets of the era. Many of the terms used to describe poetry at the time are rooted in Italian: stanza, sestina, sonetto. This is partly due to the national stereotype of Italians as being passionate and charismatic, making them ideal dramatic foils. By setting plays in a foreign country he could include political allusions in his plays while avoiding accusations of bias, both crucial at a time of high political paranoia in England. Given its popularity among poets and dramatists, what better setting than Italy?
As such, it seems likely that Shakespeare would have been surrounded by people who were versed in Italian literature and language, due to its popularity. And he'd have been talking to them to get the necessary details for the Italian settings of his plays. Dante's work was certainly known in 17th century England so it seems inconceivable that Shakespeare would not have heard about them, likely in detail, at least in second hand. The limited parallels drawn between Inferno and Much Ado About Nothing in the referenced question and answer require no more than limited knowledge of the Inferno.
That said, there is a fair degree of circumstantial evidence that Shakespeare could read Italian. It seems likely, for example, that he was well versed in French since an entire scene of Henry V is written in that language. On the presumption he did go to grammar school, he would also have learned Latin, and in the preface to an edition of the First Folio poet Ben Jonson salutes Shakespeare in spite of his "small Latin and lesse Greeke". So he seems to have had some skill in at least two languages other than English, both of which are structurally similar to Italian.
Another clue is that many of Shakespeare's plays are inspired by Italian originals. A sub-plot in The Taming of the Shrew, for example, seems to have been based on I Suppositi by Ludovico Ariosto. This is further evidence that, whether he could speak Italian or not, he was certainly familiar with its cultural influence. But we can go further. Othello follows the plot of Un Capitano Moro by Cinthio and there is compelling evidence that Shakespeare took his inspiration from the original source. There was no English translation of this available at Shakespeare's time and while a French translation existed, the structural and linguistic similarities are much closer between Othello and the original Italian.
Finally, it seems that Shakespeare, a man of relatively common birth, aspired to be seen as a contemporary gentleman. He was active in ensuring his father was given a coat of arms which he himself inherited in 1601. In The Merchant of Venice, Shakespeare has Portia complain of her suitor that:
He hath neither Latin, French, nor Italian, and you will come into the court and swear that I have a poor pennyworth in the English.
Which suggests that Shakespeare felt a familiarity with all three was a necessary accomplishment for an English gentleman. If he himself could not boast those skills, writing this line would open him up to the mockery of his peers.
- Richard Paul Roe. The Shakespeare Guide to Italy: Retracing the Bard's Unknown Travels. HarperCollins, 2011.
- Laura Tosi, Shaul Bassi (editors). Visions of Venice in Shakespeare. Routledge, 2019.
- Werner P. Friederich. Dante's Fame Abroad. University of North Carolina Press, 1950
- Naseeb Shaheen. "Shakespeare's Knowledge of Italian". Shakespeare Survey 47 (1994).