Christian Solidarity Worldwide (CSW) said Emmanuel Yohana, his wife Katherine Emmanuel and a woman identified as Khadija were arrested on 16 June at the couple's home in Zanzibar while the women were frying fish in the kitchen.
According to CSW, the Christians were informed that they had allegedly breached the law by cooking food during Ramadan. The police officer reportedly verbally abused them and allegedly said, "Today you will know how to fast".
They were released three days later following interventions by local church leaders.
CSW's Chief Executive Mervyn Thomas said, "These arrests were unjust and unwarranted. Not only were these Christians preparing food in a private home, but as non-Muslims they are under no obligation to observe the Ramadan fast. "
"Moreover, their arrests are in violation of provisions within Tanzania's constitution that recognise the right to freedom of religion or belief and prohibit discrimination on the basis of religion."
CSW said Christians are also discriminated against by being prevented from building churches because local officials impose extra-legal processes.
For example, they are often required to obtain the permission of the community surrounding the proposed site for a church before construction can begin. Due to the religious demographic being mostly Muslim in Zanzibar, local communities generally deny the requests.
Thomas added, "We also urge the Government of Zanzibar to review local planning decisions and to bring an end to the discriminatory practices, which have no basis in law and prevent Christians from building houses of worship.
"Mindful of Tanzania's religious diversity, we urge the Government of Tanzania to prioritise the respect, promotion and protection of the right to freedom of religion or belief throughout the nation for all its citizens, regardless of their creed."