Elderly priest Joseph Anselmo Mwagambwa was attacked as he was leaving an internet cafe in the island’s old town.
It follows a similar attack on two young British women there last month.
Tensions between the majority Muslim population and Christians have been on the increase in recent years, as well as on mainland Tanzania.
“He sustained burns in his face and shoulders. The acid burnt through his shirt,” Zanzibar police spokesman Mohamed Mhina told Reuters.
Tanzanian police say they are searching for witnesses to the attack which occurred in the old part of Zanzibar City, Stone Town, on Friday afternoon.
It is the latest in a series of assaults on religious figures in the country and the fifth acid attack since November, when a Muslim cleric was hospitalised with acid burns.
In a sign of further tension, a Catholic priest was shot dead in February.
The attack on the British girls in August occurred in the same part of Stone Town.
Zanzibar’s President Ali Mohammed Shein said the assault had “brought chaos and confusion to our country and outside”.
Zanzibari officials offered a £4,000 ($6,000) reward for information leading to the arrest of the suspects.
A popular tourist destination, the acid attacks came as a shock to many residents of Zanzibar who say attacks on foreign travellers are rare.
Police say no suspects have been arrested over the attack on the priest.