The answer to this question goes farther back than the Ottoman Empire.
Before the Ottoman Empire, back when The Roman Empire was still at its peak, Rome clashed and was impeded in the East by the Parthian Empire, which was removed by the Sassanian Empire. The clash of these Empires is significant in that, whenever empires conquered regions, they would seek to spread its religions and language so as to syncretize the local population into the Empire. To speak the language of an Empire was to be a part of it. As such, languages such as Greek and Latin were perceived by Middle Eastern Empires as "Roman" and later "Byzantine."
When Islam arose and began to form its own Empire, a part of their conquest was the spread of Arabic as its primary language. A correction somewhat to the original question is that, The official language of the Ottoman Empire wasn't Persian, but rather Turkish. This isn't surprising considering the Capital rested in Modern Turkey in now Istanbul. But Persian was a significant language, especially in universities and academia. The primary reason for this also stems from before the Ottoman Empire, back when Persian universities translated ancient Greek manuscripts of the Socratic Philosophers and began to write their own philosophical works in their native language, Persian.
Thus, the dismissal of the Greek language stemmed from the dominance of Semitic languages in the Middle East among the various Empires because Greek was perceived to be European/Roman. To maintain the distinct traditions and customs of the Empire included an adoption of its family of languages.