The Metropolitan Police said the men - two aged 49 and the rest aged 19, 46 and 55 - handed themselves in at a south London police station at 10pm on Monday.
They were arrested on suspicion of a public order offence and taken into custody.
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Footage earlier emerged online showing a large flammable model marked "Grenfell Tower", complete with paper figures at the windows, being set on fire.
Prime Minister Theresa May condemned the "utterly unacceptable" video, while Khadijah Mamudu, whose mother and younger brother escaped the fire, said the burning of the effigy was a "vile act".
Metropolitan Police Commander Stuart Cundy, who is leading the investigation into theGrenfell Tower tragedy, said he was "frankly appalled by the callous nature" of the video.
London Fire Commissioner Dany Cotton, who was at the scene of the harrowing blaze, said the video was "appalling and disturbing".
A total of 72 people perished as a result of the tragedy at the west London block on June 14 last year.
In the clip posted online, raucous laughter can be heard off-camera as the effigy is set alight.
A person can be heard saying: "Didn't it start from the tenth floor, though?", while others add: "Help me! Help me!" and "Jump out the window!"
Another says: "Here we go," as the fire takes hold.
At the end of the clip, someone else can be heard saying: "That's what happens when they don't pay their rent."
The video ends as the model is completely consumed by the fire, with one bystander saying: "Perfect."
Campaigners described the video as a "sickening act of hate".
Natasha Elcock, from Grenfell United, said: "It's a disgusting video. We hope that the police are taking this seriously", while Justice4Grenfell said it "caused great alarm and distress".
London mayor Sadiq Khan said he "utterly" condemned the "sickening" video, describing the Grenfell Tower fire as "one of the most devastating tragedies our city has ever suffered".