The angry protesters barged their way through an automatic door at Kensington and Chelsea town hall and sought to gain entry to an upper floor.
The deadly blaze has mushroomed into a political crisis, testing the fragile government of Prime Minister Theresa May.
But the Conservative leader still struggled to overcome accusations that she lacked compassion because she had failed to meet with victims on her first visit to the devastated site.
Residents had raised concerns about the fire safety of the building prior to the blaze.
He added: "I have said it before, you have my absolute assurance that as soon as I can possibly tell you something that I know to be accurate, I will tell you".
Asked repeatedly whether she had misread the public mood, May did not answer directly but said the focus was on providing support to the victims.
Details were not disclosed, but an unidentified group spokesman said members had given May their "demands and expectations" and that a full statement would be made only "in the community, with the community".
Grief and disbelief at seeing a tower that had been home to up to 600 people reduced to a burned-out shell turned to outrage and anger. "I really hope it won't, but it may increase", he said.
The country will fall silent as the nation remembers all who lost their lived in Wednesday's tragedy as well as thinking about all the survivors who lost their homes in the blaze.
"But it is a bad tragedy".
Metropolitan Police said family liaison officers have been deployed to support 52 families of residents who died, were injured or were reported missing. As an absolute priority, what we are all doing is, as quickly and with as much dignity as we can, recovering those that are still inside.
The fire has also become a symbol of class inequality - the Grenfell Tower's charred remains stand high above one of the British capital's wealthiest neighborhoods.
He said: "Clearly it's quite unprecedented for 16 very ordinary people, who this time last week were walking their dogs or talking to each other in the streets around here and north Kensington, actually to be in 10 Downing Street talking face-to-face with the Prime Minister".
"We have identified those buildings and now and over the weekend people are going in and inspecting those buildings".
"Put to the test, the United Kingdom has been resolute in the face of adversity".
This year, however, it is hard to escape a very sombre national mood.