They must stay in fixed places allocated to them by the government and not travel elsewhere, police say.
Bangladesh also announced plans to build shelters for up to 400,000 people near the city of Cox's Bazar.
The mainly Muslim Rohingyas have been fleeing a Myanmar government offensive since last month, which the UN says could amount to ethnic cleansing.
But the army says it is responding to attacks by militants and denies it is targeting civilians.
Meanwhile, Amnesty International says it can reveal new evidence pointing to a mass-scale scorched-earth campaign across northern Rakhine State, where Myanmar security forces and vigilante mobs are burning down entire Rohingya villages and shooting people at random as they try to flee.
The AI says the organization’s analysis of active fire-detection data, satellite imagery, photographs and videos from the ground, as well as interviews with dozens of eyewitnesses in Myanmar and across the border in Bangladesh, shows how an orchestrated campaign of systematic burnings has targeted Rohingya villages across northern Rakhine State for almost three weeks.
The Rohingya, a stateless mostly Muslim minority in Buddhist-majority Rakhine, have long experienced persecution in Myanmar, which says they are illegal immigrants.
The Sri Lankan government is yet to make an official stand known with regard to this issue.
Rohingyas crossing the Naf river
Foreign affairs state minister Wasantha Senanayake has told BBC Sinhala Service that Sri Lankan’s stand was that all countries should act democratically and ensure the rights of all their citizens.
Don’t isolate Myanmar
As a friendly country, Sri Lanka wants Myanmar not to be isolated internationally over this matter, and the way out is through negoations, he said.
Sufferings of women & children
Asked about his comments as to why Aung San Suu Kyi is not acting over this issue, he said she was yet to have full powers, although she was a heroine who worked for her country’s independence.
The US secretary of state has urged Myanmar to treat all its citizens equally, while the UN secretary general condemned the violence there.
Senanayake said Sri Lanka agreed with their demand for an immediate halt to the violence, and reiterated that international-isolation would turn countries violent, taking North Korea as a case in point.
He said there could be ‘hidden forces’ behind the Myanmar crisis.
In Colombo and at Katankudy, Oddamawadi, Akkarapattu and Kinniya in the eastern province, protests took place against attacks of the Rohingyas.
Around 30 Rohingyas who fled Myanmar arrived in Sri Lanka and are being kept under UNHCR care at a house at Mt. Lavinia following a court order, said their lawyer Shiraz Noordeen.
Protests at Katankudy, Oddamawadi, Akkarapattu and Kinniya
Noordeen said the Sri Lankan foreign ministry has signed an expression of interest with the UN with regard to these people.
He expressed hope the UN would send the Rohingya people to another country, in the same manner it acted when 138 Rohingyas came to the island two years ago.
State minister Senanayake said they could stay in Sri Lanka until the UN took a decision.