The Polish-Lithuanian state was multi-confessional since its inception in 1386 (a personal union resulting from the marriage of Jogaila and Jadwiga). Although Catholic, it had a very large and powerful Eastern Christian minority, along with a Jewish community that enjoyed state protection and legal exemptions. As such, there was a large degree of cultural "normalization" in the presence of yet another confessional group like the Islamic Tatars. According to a mid-sixteenth century anonymous Islamic historian, they arrived in Lithuania sometime in the fourteenth century (during the reign of Timur) and shortly thereafter came into the employ of the Polish crown as auxiliary soldiers. There is some evidence that Grand Duke of Lithuania Vytautas (+1430) also utilized Islamic Tatars as part of his military. Prowess in battle, in turn, won them considerable social currency. Being recipients of royal land grants for bravery, their military leadership class had all the social respect of the Christian nobility, though without any actual political sway, as they were barred from parliamentary participation or reception of royal offices. In instances of conversion to Catholicism, however, they were immediately ennobled. In addition to royal favor and social currency that came with military valor, especially post-Reformation, the discourse regarding Islamic Tatars was that they were confessionally benign, ie, they were not actively seeking converts, unlike the Lutheran, Calvinist and Anti-Trinitarian confessional "other."
As a final note - not all Islamic Tatars in Lithuania were soldiers - many were artisans, especially esteemed for their production of leather goods. A significant portion resided in small Tatar villages, but many also lived in larger Lithuanian urban centers, like Vilnius and Trakai, where they had their own mosques (always just outside city walls) and rubbed shoulders daily with their Christian and Jewish neighbors.

I cannot think of any English language literature on the topic, here are a few Polish secondary materials: Piotr Borawski; Aleksander Dubiński, Tatarzy polscy dzieje, obrzędy, legendy, tradycje, Warszawa: Iskry, 1986; Piotr Borawski, Tatarzy w dawnej Rzeczypospolitej, Warszawa: Ludowa Spółdzielnia Wydawnicza, 1986; Vytautas Ališauskas, Liudas Jovaiša, Mindaugas Paknys, Rimvydas Petrauskas, Eligijus Raila, Kultura Wielkiego Księstwa Litewskiego. Analizy i obrazy, Kraków: Universitas, 2011 (section on Tatars, pgs 760-771).

EDIT: Closest thing in English I can think of that even addresses Islamic Tatar communities in the region is David Frick's recent work on seventeenth century Vilnius: David Frick, Kith, Kin, and Neighbors: Communities and Confessions in Seventeenth-Century Wilno, Ithaca, NY: Cornell University Press, 2013.