Government security forces formed a barricade Saturday to control a demonstration near the church, but several people broke through and set fire to the church, AllAfrica.com reported.
The church property also has a seniors' care facility, student housing, classrooms and a medical clinic, the Sudan Tribune reported.
Tensions along the border between Sudan and South Sudan have been mounting since last July when the predominantly Christian South Sudan seceded and gained independence from the mostly Muslim north.
This month, the Sudanese government announced more than 500,000 residents from South Sudan were considered foreigners without full citizens' rights.
Fundamentalist Muslims are also demanding the deportation of people from South Sudan, the Tribune said.