Solihull Council said an investigation would take place after a complaint about the post by Conservative Councillor Ken Hawkins.
The tweet, replying to an image of Kensington Town Hall protesters posted by Guardian journalist Martin Belam, read: "Lets get ourselves a hangin!"
Mr Hawkins has apologised "unreservedly".
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The tweet, which was posted on Friday, has since been deleted.
It came after protests were held as residents demanded support for those affected by the Grenfell Tower fire, and more information about what happened.
Council leader Bob Sleigh said he had taken "immediate action" to suspend Mr Hawkins and would take over his responsibilities as cabinet member for the environment, housing and regeneration in the interim.
"In light of the complaint related to Cllr Hawkins' comments on social media, an investigation will now take place led by the council's monitoring officer in line with the council's standards procedure," he said.
In an apology posted to Twitter on Monday, Mr Hawkins said: "I fully appreciate that my tweet on Friday which related to the protest outside and inside the Royal Borough of Kensington & Chelsea council offices may have caused offence and I unreservedly apologise for that.
"In no way whatsoever was the short message directed at the unfortunate victims of the Grenfell Tower fire and I am distraught that some people have interpreted it in this way."
A councillor has been suspended from his cabinet position after an offensive tweet about Grenfell Tower protesters.