First of all, insofar as we can speak of Islam as having a formal language policy at all, that policy would be defined by the extensive body of jurisprudence on which languages may or must be used in particular religious contexts. Ṣalāt, ritual prayer, has to be in Arabic (Mawdudi 1957 notes a few arguable exceptions to this). Duʕā', asking favours of God, may be in any language. The adhān, the call to prayer, has to be in Arabic according to most scholars, although Atatürk briefly forced Turkish mosques to make it in Turkish (Atalay 2012). For the khuṭbah, the Friday sermon, scholars' opinions differ – to keep on the safe side, it's common for the imam to deliver a sermon in the congregation's language followed by a much shorter sermon in Arabic. The Qur'ān may be translated, and since early times frequently has been, but no translation of it can be considered authoritative, or substituted for the original in ritual contexts; in fact, such translations are viewed more as commentaries than as versions of the original. Everyday religious formulae – bismillah (in the name of God), alhamdulillah (thank God), inshallah (if God wills), etc – are ordinarily in Arabic, though I don't know what the jurists have to say about that.
As a result, the ordinary believer is commonly exposed to Arabic in religious contexts, and is individually required to memorise a certain number of formulae and chapters of the Qur'ān in Arabic. Quite frequently, the latter in particular are learnt by heart early with only cursory explanation of their meaning, since reciting them verbatim is a precondition for proper prayer, but understanding them is only really vital at a more advanced stage. What does need to be understood immediately – the basic religious obligations, creed, etc – is explained in a language the student understands. However, the further a student advances, the more important it becomes to have direct access to the original source texts; thus learning Classical Arabic is a basic prerequisite for becoming a serious religious scholar, although the vast majority of Muslims never get that far, and indeed a majority of Muslims do not speak Arabic. Regionally, other languages may also come to assume a secondary position in religious education – for example, Urdu in Pakistan, even though most students there have a different first language. A remarkable example of this is to be found in northeastern Nigeria, where advanced religious education requires mastering not just Classical Arabic but also Classical Kanembu, an extremely archaic variety of Kanembu currently used only for explaining Classical Arabic texts (Bondarev & Tijani 2013).
Interpreting the notion of "language policy" more broadly, one might also talk about the influence of Islam on attitudes to language. In this connection, the obvious point to discuss would be the (very weakly supported) claim commonly heard that "Arabic is the language of Paradise", and the even more obviously fabricated claim sometimes heard east of Iraq that "Arabic and Persian are the languages of Paradise". Yet the weakness of the religious evidence for both assertions is a strong indication that the causality is the other way around: religious positions on language, in Islam as elsewhere, have often been influenced by extra-religious prejudices. The universal consensus that some Islamic rituals must be performed in Arabic make it difficult for any Islamic society to assert strongly negative attitudes to Arabic, but beyond that minimum, language attitudes are determined more by social and political factors than by Islam specifically.