Kurds straddling Kermanshah and Ilam provinces, who speak a language of their own, are Twelver Shia Muslims (as you perhaps know, what is referred to as Kurdish in fact consists of several mutually unintelligible western Iranian idioms).
So are the Laki, who reside on both sides of the border between Iran and Iraq (where the Faylis, many of whom Saddam expulsed to Iran, speak a dialect of Laki). It could be argued that their language is closer to Lori (Lors represent another major linguistic group in central-western Iran), although it is generally classified as a Kurdish one.
Also some 15% of eastern Turkey's 3 to 4 million Zaza (usually designated as or associated with the Kurds) adhere to Alevism, an unorthodox branch of Shiaism (the rest of the Zazas being Sunni Muslim). While this is not a particularly important percentage, it is nonetheless the highest among any linguistic group (the percentage of Alevis among native Turkish-speakers for instance is far less).