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David Buttress, the founder of online food order service Just Eat, was appointed to help support people with rising bills and advise on ways to boost the faltering economy.
His new role was announced by Downing Street on Tuesday.
But in a series of tweets, it was revealed Mr Buttress had said his new boss “had to go” over the partygate scandal.
In January, he tweeted: “Why is it that the worst people often rise to the highest office and stay there?
“For me it isn’t important what job you do or your title, but it is vitally important why you do the job and what you achieve.
“Boris has to go, he has to. You can’t survive judgement like this.”
In another, he attacked Mr Johnson's private education and said: "Does anyone believe he would be PM on merit? Not a chance mate. 19 PMs from Eton."
He also said the Government’s “arrogance and ignorance” was “breathtaking” when cabinet minister Jacob Rees Mogg defended the Prime Minister.
Mr Buttress also accused Conservative MPs of neglecting of Wales and has called for the country to be independent of Westminster.
Responding to a tweet from Welsh Conservative leader, Andrew Davies, claiming workers in Wales take home £3,120 a year less than people in Scotland because of Labour policies, Mr Buttress replied: “How on earth can the party of Thatcher have the audacity to tweet this!?
“This blows my mind. Destroyed Welsh communities and the concept of society in England.”
In another tweet, he said: “Wales will be a huge success as an independent country economically, culturally and community wise.
“I’d bet my life on it. Being part of Westminster is a disaster for Wales, decades and centuries of lack of respect, investment and focus.
“In the end it is as simple as that.”
The former Just Eat CEO was awarded the cost of living role by the Chancellor of the Duchy of Lancaster Steve Barclay on a voluntary basis.
The role involves working with the private sector to identify, develop and promote business initiatives that support will people with the soaring costs of living, the government said.
It includes building on price locking campaigns, product offers and the expansion of business programmes.
He is set to hold the position until the end of the year.