usage - Partisans interest - meaning

This is probably a typo for partisan interest. A "partisan interest" is an interest that derives from one's affiliation with a party or faction.

In this context, the sentence you've quoted is basically saying that Muslims should prioritize the interests of Islam as a whole over the interests of subgroups of Islam or other factions (e.g. the interests of Shias, Sunnis, Afghans, Arabs, etc.).