Authorities have banned gatherings of more than four people in the most tense northern parts of the state while police have issued warnings about "provocative messages" spreading on social media.
Tripura has a 850-kilometre (525 mile) long border with Muslim-majority Bangladesh, where seven people were killed when a mob ransacked a Hindu temple this month.
The riots, sparked by footage of a Quran being placed on the knee of a Hindu god during celebrations for a Hindu festival, spread to 12 districts in Bangladesh.
India has expressed concern at the attacks and troubles erupted in Tripura on the sidelines of a rally of hundreds of followers of the right-wing Vishwa Hindu Parishad group.
At least four mosques were vandalized and shops and homes owned by Muslims were ransacked.
Tripura is ruled by the Bharatiya Janata Party of Prime Minister Narendra Modi.
Leaders of India's minority Muslim community say they have increasingly faced attacks since the Hindu nationalist party came to power in 2014.