There are more Muslims in the Divine Comedy than just Avicenna. Averroes, Siger of Brabant, and Saladin come in for a share of honor. Let us not forget that after the Muslim conquest there was an Islamic presence in Europe from the 8th century to the 15th, particularly in Spain. Venice was a major sea port which had regular commerce with the Muslim world. There have been numerous papers published about the indirect influence of Islam on the Divine Comedy, including speculation that he might have borrowed certain incidents from the "Kitab al Miraj."
I think Dante himself makes an argument for Homer being the center of the canon in the Divine Comedy. Though he's gone he's not forgotten.
Just because they didn't have the text anymore doesn't mean they didn't know what was in it or the man's reputation in past ages. Petrarch even wrote a letter to Homer among his letters to famous dead people, so it shows his presence was felt, especially through Virgil. I might also remark that Ovid seems to have been more popular in the middle ages and during the Renaissance than Virgil was. To quote Petrarch himself from "An Excursion to Paris":Homer is he, the poets' sovran lord;
Next, Horace comes, the keen satirical;
Ovid the third; and Lucan afterward.
It seems as though Virgil has always been more admired than liked.But, lest you should be misled by my words, I hasten to add that there are no Virgils here, although many Ovids, so that you would say that the latter author was justified in his reliance upon his genius or the affection of posterity, when he placed at the end of his Metamorphoses that audacious prophecy where he ventures to claim that as far as the power of Rome shall extend, - nay, as far as the very name of Roman shall penetrate in a conquered world, - so widely shall his works be read by enthusiastic admirers.
I've found that most well read people seem to be aware of Firdawsi and the Mahabharata at least since the late 1700s. There was a flood of interest around the time that Goethe was imitating Hafiz with his West-Eastern Divan. This fascination with the far away and exotic seems to be an inspiration for such Enlightenment writings as Montesquieu's Persian Letters(1721). Sir William Jones takes up the torch later in the century translating all manner of Persian, Sanskrit, and Mandarin works, really doing some pioneering work in the area of Oriental Studies. However, there had been small chairs in language departments for Oriental Studies going back to the Renaissance, and I know that Christian missionaries were in those regions since at least the late 1500s. So you can't say we completely ignored them until the 19th century.
I don't know that you can claim there was no interest in Japanese or Eastern literature before the 40's. Ezra Pound came out with his translation of Li Po "The River Merchant's Wife: A Letter" in 1915. He tried to translate some other Chinese poems, with varying success. Arthur Waley made a prominent career out of translating Chinese and Japanese literature around the same time. He publishes The Tale of Genji in the 20s and 30s. The Harvard Classics came out in 1910 and included the works of Confucius, the Koran, and the Bhagavad-Gita. The Chinese play The Orphan of Chao was popular in Voltaire's day. Arthur and Edmond Warner translated the complete Shahnameh into verse in 1905. Pearl S. Buck, a Nobel Prize winner, translated Water Margin into English in 1933. Rabindranath Tagor, an Indian, won the Nobel Prize in 1913. Ganguli translated the complete Mahabharata into English in 1896. Goethe talks about how much he admired Kalidasa, and you can see the influence of Sakuntala and the Ring of Recollection on his Faust.
Before the 19th century, can you see the influence of European literature on China? What impact did Homer, Shakespeare, Dante, or Goethe make in those circles?