Martin Hewitt, chairman of the National Police Chief Council, said people should think carefully to avoid inciting others to violence.
It comes as it was revealed that police have 10,000 officers ready to deploy at 24 hours’ notice as part of possible no deal Brexit preparations.
There have been rising tensions in recent weeks with a number of demonstrations and protests over Britain’s failure to leave the EU on time on March 29.
However, Mr Hewitt said the readying of officers was a precaution and that the NPCC was preparing for the "worst case scenario" and was not predicting major problems.
Chief Constable Charlie Hall, the NPCC lead for operations, also said there was no intelligence to suggest there would be a rise in crime or disorder because of Brexit, although forces were "prepared to respond to any issues that may arise".
Mr Hewitt said the UK was in "an incredibly febrile atmosphere" as a result of the debate over leaving the EU and there was a lot of "angry talk" on social media – much of it aimed at MPs, the media and major organisations.
He said: "I think there is a responsibility on those individuals that have a platform and have a voice to communicate in a way that is temperate and is not in any way going to inflame people's views."
Officers in charge of policing Parliament said they had seen an increase in abuse aimed at politicians and several MPs have requested increased security.
Only a small number of crimes have been linked directly to Brexit, police said, with about half being malicious communications, while the rest included verbal abuse, harassment and offences committed during protests.
But hate crimes remain higher than before the 2016 EU referendum.
After warnings of disruptions at the border and to food supply chains if the UK leaves without a deal, police said they had plans to deal with incidents such as problems on the roads, major protests or even rioting and looting.