Post Publishing did not detail his decision to step down from the board.
Mr Wissanu also resigned as board chairman of three listed firms - Bangkok First Investment & Trust, Sikarin and Millcon Steel - on Friday.
He told Thai Rath TV that he had been approached to join the government led by Prime Minister Prayuth Chan-ocha as a deputy prime minister in charge of legal affairs.
Mr Wissanu was a key legal expert in the drafting of the interim charter that took effect on July 22.
His resignation from Post Publishing followed the departure of Pridiyathorn Devakula, who stepped down from the board on Thursday.
The former Bank of Thailand governor is widely tipped to be a deputy prime minister overseeing economic policy for the government. He is chief economic adviser of the National Council for Peace and Order.
Post Publishing's publications include the <i>Bangkok Post</i>,<i> Post Today</i> and M2F.