Under the new law, a blogger can face fines of up to $2,200 for publishing content considered "indecent, obscene (or) hate speech", or even just for causing "annoyance".
The law has fuelled criticism of President John Magufuli.
He took office in 2015 as a corruption-fighting "man of the people".
But he has earned criticism for his authoritarian leadership style, with detractors saying he has clamped down on opposition and freedom of expression.
Reporters Without Borders (RSF) on Tuesday called on Tanzania to scrap a new law regulating Internet content, saying it targeted critics who had sought a "safe haven" online.
The new law requires the operators of online platforms to divulge the names of their sources and contributors if the authorities require it.
News websites and blogs that are in breach of the law could be banned from posting new content and face prosecution.
Refusing to comply with this requirement by a deadline of Friday, Jamii Forums, the country's biggest online news site, which has more than 526,000 followers on Twitter, has suspended publishing, RSF said.
The law also requires online platforms to pay for a licence and other fees that amount to around $900 (750 euros) euros a year — failure to comply may result in closure, a fine of up to $2,200 and a 12-month jail term.
"Some online information providers are going to disappear because they cannot afford the exorbitant cost of a licence, while others, such as Jamii Forums, may have to cease operating in Tanzania for good because the authorities want them to reveal their sources," said RSF's head of Africa desk Arnaud Froger.