On Saturday, protesters blocked motorways and attacked trains and stations in West Bengal's capital.
A curfew was also imposed in Guwahati, in the north-eastern state Assam, after two people died in clashes this week.
The curfew was lifted from 09:00 local time (03:30 GMT) until 16:00 (10:30 GMT) on Saturday. Protest groups in Guwahati have said they plan to defy the curfew on Saturday evening.
The UK, US and Canada have issued travel warnings for people visiting India's north-east, telling their citizens to "exercise caution" if travelling to the region.
The new law entitles non-Muslim migrants from three Muslim-majority countries to citizenship if they are facing religious persecution.
On Saturday, protesters blocked motorways and attacked trains and stations in West Bengal's capital.
A curfew was also imposed in Guwahati, in the north-eastern state Assam, after two people died in clashes this week.
The curfew was lifted from 09:00 local time (03:30 GMT) until 16:00 (10:30 GMT) on Saturday. Protest groups in Guwahati have said they plan to defy the curfew on Saturday evening.
The UK, US and Canada have issued travel warnings for people visiting India's north-east, telling their citizens to "exercise caution" if travelling to the region.
Muslim rights groups across the country and an opposition political party argue the bill is part of Prime Minister Narendra Modi's "Hindutva", or Hindu nationalism, and part of an agenda to marginalise Muslims.
Mr Modi denies this, saying that Muslims are not covered by the new law because they are not religious minorities, and therefore do not need India's protection.
Critics argue that if the law were genuinely aimed at protecting minorities, it would include Muslim religious minorities who have faced persecution in their own countries - Ahmadis in Pakistan, for example.
They also say the bill violates secular principles enshrined in the constitution, which prohibits religious discrimination against all citizens.
There is also an anti-migrant element to some of the protests.
The BBC's India correspondent, Soutik Biswas, said that the protests in Assam in particular have little to do with the law being seen as exclusionary or as a threat to secularism, and "more to do with indigenous fears about being demographically and culturally swamped by 'outsiders'".
Prime Minister Modi has sought to reassure people in Assam, telling them they had "nothing to worry" about.
"The central government and I are totally committed to constitutionally safeguard the political, linguistic, cultural and land rights of the Assamese people," he tweeted on Thursday.
But it is unlikely residents would have been able to read the message, because internet and mobile services in the area were shut down.
Has the bill been legally challenged?
The Indian Union Muslim League, a political party, has petitioned the country's top court to declare the bill illegal.
In their petition to the Supreme Court, the Indian Union Muslim League argued that the bill violated articles of equality, fundamental rights and the right to life.
More than 700 eminent Indian personalities, including jurists, lawyers, academics and actors, have signed a statement "categorically" condemning the bill.
Investigative journalist Rana Ayyub told the BBC that it was sending out the wrong message.
"Clearly you are catering to your Hindu base by telling them that this country is only for Hindus," she said.
"The world's biggest democracy had a big heart when it could accommodate people. Right now we are coming across to the world as petty vindictive civilisation. That's not what India stood for."