The men, two aged 49 and the others 55, 46 and 19, have been taken into custody by police.
Earlier, Scotland Yard has confirmed that five men presented themselves to a police station after an investigation into the video has been launched.
The group placed the effigy on top of the bonfire. As the flames consumed it, people in the group chanted "help me, help me." In the video they laughed and cheered, underscoring that this was an attempt at black humor rather than a considerate act of protest.
UK Prime Minister Theresa May was among those offended; she tweeted: “To disrespect those who lost their lives at Grenfell Tower, as well as their families and loved ones, is utterly unacceptable.”
Soon after the video was posted, the police launched an investigation into it.