BAGHDAD—Islamic State’s rise is dividing Iraq’s Sunni minority, pitting tribes loyal to the militant group against those who support the government and, in some cases, cleaving tribes apart from within.

Sheikh Dhaher Bedewi, a tribal leader in Anbar province, said that while in the mid-2000s his fighters handed captured Sunni insurgents over to U.S. forces, mercy is no longer an option.

“This...