The Taiping Rebellion, and the Albigensian Crusade, for instance. Whether there's any current civil war that is primarily caused by religion, I'm not sure. Arguably, religion started the current conflict in Yemen:
According to Human Rights Watch, an armed conflict between Yemeni government forces and Huthi rebels began in 2004. Husain al-Huthi founded the Believing Youth movement in the 1990s, aimed at reviving Zaidi Islam, a branch of Shi’ism found mainly in Yemen, to counter growing fundamentalist Sunni trends in the northern Yemeni governorates where Zaidis dominate. The conflict began as isolated clashes between the Believing Youth movement (Huthis) and the army in Sa’da. Thereafter, anti-Israel and anti-US demonstrations led by Huthis in San’a, Yemen’s capital, which embarrassed the government after it had embraced US counter-terrorism efforts, led to arrests of Huthis and further clashes with them.
Zaidi Hashemites, descendants of the Prophet Muhammad, led the Huthi movement. They had ruled Yemen for a millennium and comprised the state’s religious and governing elite until the army-led revolution in 1962, also supported by some Zaidi tribes, deposed them. Zaidi Hashemites are especially prominent in the Sa’da area, where there has not historically been a significant government law enforcement presence.
...
The Yemeni Government has admitted the presence of Al-Qaeda within its borders, placing the number of members at 300 in 2009. In January 2010, under international pressure, Yemen declared "open war" against the south-operating armed group. The US has used drones in operations against Al-Qaeda, which has resulted in such high-profile killings as that of Anwar al-Awlaki leader in September 2011. Drone attacks have continued into 2012. According to the Ploughshares Website much of southern Yemen, especially Abyan Province, was captured by AQAP in the first half of 2011, and AQAP gained control of the city of Jaar in March 2011.
http://www.geneva-academy.ch/RULAC/current_conflict.php?id_state=234
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