During the medieval period (and the fifteenth century—when the real, live Fra Lippo Lippi was kickin' around—certainly qualifies as medieval), Latin was the language of the upper-crust. Anything of permanent value was written in Latin. It was the language of the Church and of the highly-educated. In the poem, then, Latin is a symbol for not only the Church itself, but also education and the cultural superiority that goes along with it.
Line 109: Since Lippo is a monk by necessity and not by choice, he lets us in on the fact that he's not exactly bookish, and teaching him Latin would have been a complete waste of time. So, he's set apart from the typical educated monks.
Line 111: All the Latin Lippo cares about is how the verb "amo" (I love) is conjugated. It's fitting that the speaker expresses this here in a snippet of popular melody of the time—basically, a love song.
Line 242: Lippo uses the reference to Latin here to emphasize how the clergy thinks they are better and know better than anyone else. In particular, he's angry at how they set themselves up as authorities on art and their anger at how he's spending too much time making his paintings too realistic, not spiritual enough.
Line 281: Latin here serves as a class marker. It separates the hoity-toities from the street-level peeps, such as the guardsman. Lippo uses the fact that neither he nor the guardsman speak Latin to put them on the same level class-wise. It's a rhetorical technique Lippo uses to charm the guardsman and get him on his side.