I can almost certainly guarantee you that Dante Alighieri never once wore a laurel wreath in his life. The image we all recognize of Dante Alighieri (complete with laurel wreath) was painted by Sandro Botticelli over a hundred years after his death. Most successive depictions of Dante, including the one on the Italian two-Euro coin, are based on this image.

The laurel wreath isn't an allusion to ancient Rome: the poet/writer Francesco Petrarca (Eng.: Petrarch) had been crowned with a laurel wreath by the Pope in 1341; the ancient roman tradition had been revived for the the occasion of recognizing Petrarch for his literary work, and consequentially subsequent depictions of Petrarca feature him wearing a laurel wreath, even if we have no indication that he kept wearing it after his "coronation". In the following centuries, others, like writer/poet Torquatio Tasso, would also earn a "Laurel-Wreath Coronation", and to this day the tradition is carried out by the President of the United States who nominates a "Poet Laureate", and it is also tradition for Italian students to wear a laurel wreath after graduation to signify a great intellectual/academic achievement.

As Dante Alighieri had rather rapidly entered the Italian popular imagination as "The Divine Poet" (Boccaccio was writing very highly of him in 1346, over twenty years after his death) it didn't take long for his image to also be associated with a laurel wreath, even if he died two decades before the tradition started!