Foreign nationals directly affected by the Grenfell Tower fire are to be allowed to stay in the UK for 12 months regardless of their immigration status.
The Home Office said it would not conduct immigration checks on survivors and those coming forward with information.
Labour called for a wider "amnesty".
Meanwhile, ministers have ordered a taskforce to help run Kensington and Chelsea Council, which has faced heavy criticism for its handling of disaster.
The specialist team will take over the running of key services, including housing and the longer term recovery of the area in North Kensington.
At least 80 people died in the fire on 14 June.
The Home Office said its priority was to see residents "deal with the extremely difficult circumstances" so they could start to rebuild their lives.
In a written statement to Parliament, Home Office minister Brandon Lewis said: "Everyone affected by this tragedy needs reassurance that the government is there for them at this terrible time and we will continue to provide the support they need to help them through the difficult days, weeks and months to come."
He said extending the period of leave to remain for foreign residents affected by the fire would also allow them to assist the police and other authorities with their inquiries.
Shadow home secretary Diane Abbott said the government should give permanent residency to the residents.
"Some survivors have literally lost everything in this horrific tragedy, all their possessions, homes and loved ones," she said.
"The idea that on top of this they could be deported later is grotesque."
A statement from the Met Police said 250 specialist investigators were working on the inquiry into the fire and the last visible human remains were removed from Grenfell Tower on Monday.
Met Police Commander Stuart Cundy said there had been a total of 87 "recoveries" but, due to the "catastrophic damage" inside, that did not mean 87 people.
So far, 21 people have been formally identified and their families informed.
More inquests into the deaths of victims have been opened, with the Westminster coroner hearing the body of one of the oldest people to have been killed was identified by dental records.
Dr Fiona Wilcox was told the body of 84-year-old Sheila, formerly known as Sheila Smith, was found on the 16th floor, while Vincent Chiejina, 60, was recovered from the 17th floor and identified by DNA.
Earlier, housing minister Alok Sharma fought back tears as he told the Commons of hearing "harrowing accounts" from survivors, saying it had been the most "humbling and moving experience of my life".
Only 14 out of the 158 affected families have accepted offers of temporary accommodation but ministers say no-one will be forced to move.
Mr Sharma said 19 families "have not yet been ready to engage" in the process of being rehoused, while others were waiting for offers of permanent tenancy and many were still in hotels.
But he acknowledged some residents still had a "lack of trust" in the authorities.
Elizabeth Campbell, who is taking over as the new Kensington and Chelsea Council leader, denied the council was "being taken over by outside commissioners" after the government sent in a taskforce to run some of its services.
"We have asked people to come because we need more help because this is something on a national scale," she said.
"We will do absolutely everything we can as a council to help our community and to help our community heal."
At the Scene
By Frankie McCamley, BBC News
The mood is tense in the area surrounding Grenfell Tower.
Many residents have been living in small hotel rooms, with four people crammed into each room.
They are desperately trying to carry on with their lives by taking their children to school and going to work. But the stark reality is that they are not in a place they can call home.
Both adults and children are having trouble sleeping, waking up to nightmares of the tower burning. One parent explained that his daughter kept drawing pictures of the building on fire.
Despite counselling sessions on offer at local community centres, residents say they want people to visit them at their hotel.
They feel the help should be coming to them. They say they should not be going in search of help.
Many are traumatised and feel they are not being treated like victims. This is causing hostility and anger towards the services.
Many have also turned down offers of temporary accommodation.
Residents say they want to move into somewhere permanent and nearby. Many explain they have been offered numerous places that simply are not suitable due to the size, location and disabled access.
The newly-elected Labour MP for Kensington, MP Emma Dent Coad, told Mr Sharma that some residents were being offered "totally unsuitable accommodation".
The retired judge chairing the public inquiry into the fire has promised to hear from people directly affected.
Sir Martin Moore-Bick, who has faced calls to stand down, initially suggested the inquiry may not be broad enough to satisfy survivors.
Launching a consultation document, the retired judge said: "I am determined to establish the causes of the tragedy, and ensure that the appropriate lessons are learnt.
"To produce a report as quickly as possible, with clear recommendations for action, I will listen to people and consider a broad range of evidence, including on the role of the relevant public authorities and contractors, in order to help me answer the important questions."
Earlier, the government said 190 buildings in England that underwent fire tests on their cladding - a renovation that is thought to have contributed to the spread of the Grenfell Tower fire - have failed. It also announced that cladding from one building had passed the test - the only sample to do so to date.
In the afternoon, emergency teams working on the shell of Grenfell Tower were temporarily withdrawn after sensors in the building showed it had shifted more than 5mm.
The public were said to be at "no risk" and the work later restarted.
But the use of air horns to alert crews was reported to have "upset" some neighbours of nearby blocks, prompting officials to say the practice would not be repeated in future.