The president has "terminated the appointment of the Minister of State at the Ministry of Finance, Mr Charles Adu Boahen, with immediate effect," he said in a statement.
The fallout from the expose comes as the government is under pressure over Ghana's economic crisis and lawmakers push Akufo-Addo to fire Finance Minister Kenneth Ofori-Atta.
The presidency's statement said Akufo-Addo's decision came after "being made aware of the allegations" against Boahen in the documentary "Galamsey Economy."
Teasers from the expose show Boahen in what the documentary claims are images of him trying to demand $200,000 from potential investors to give to the vice president to allow them to do business.
Boahen has not commented on the allegations made in the teasers.
But before the sacking was announced, Vice President Mahamudu Bawumia on Monday said he was not aware of any meeting in which Boahen had used his name to "peddle influence and collect money from supposed investors."
"If what the minister is alleged to have said is accurately captured in the video, then his position as a minister of state is untenable," he wrote on Twitter.
"I will not allow anyone to use my name to engage in corrupt activities."
Akufo-Addo also referred the case to prosecutors for further investigation.
The Ghanian president has been under increasing pressure after recently opening negotiations with the International Monetary Fund (IMF) over a potential $3 billion loan to help shore up the country's economy.
Last month he appealed to Ghanaians to support his efforts to manage the "crisis" as inflation has hit 40 percent and the national currency, the cedi, has dropped sharply.