The video prompted a police investigation, and on Monday evening, five men between the ages of 19 and 55 were arrested after turning themselves in.
The video clip, posted online, showed a large effigy marked "Grenfell Tower", complete with paper figures at the windows, being set ablaze on a bonfire.
While Khadijah Mamudu, whose mother and younger brother escaped the fire, said the burning of the effigy was a "vile act".
A total of 72 people died in the fire, which engulfed Grenfell Tower, a west London block, on 14 June a year ago. But raucous laughter can be heard off-camera, with several bystanders speaking with southern English accents.
A video, circulated first on WhatsApp and then Facebook and Twitter, showed a gang of people laughing and joking as they placed a model of the Tower onto a bonfire.
A person can be heard saying: "Didn't it start from the tenth floor, though?", while others add: "Help me!"
'Elhouhabi shamelessly took advantage of the efforts to rehouse people whose homes had been destroyed by claiming his link to one of the flats, and he was subsequently put in hotel accommodation.
The Scotland Yard commander, Stuart Cundy, who is leading the investigation into the Grenfell fire, which claimed 72 lives, said he was "appalled by the callous nature of the video posted online".
London Fire Commissioner Dany Cotton has also voiced concern over the video, calling it appalling and adding that the actions of those involved "cannot be justified on any level".
Such was the public outcry over the video that even the Prime Minister Theresa May responded, condemning it as "utterly unacceptable".
"It can not be justified on any level".
One person shouts "jump out the window" while another offers the original fire brigade advice before the scale of the tragedy was understood, saying "stay in your flats".
Some partygoers were aware of this: at least one could be heard commenting the bonfire was "really bad taste".
Grenfell United, the official organisation for bereaved, survivors and community of Grenfell Tower, said: "It's a disgusting video. I urge social media companies to do the right thing and remove this content immediately".
MEND, an anti-Islamophobia organization, said the "sickening Islamophobic" video amounted to a hate crime.