The remarks came after 10 police officers visited the BBC's office in Bangkok.
Deputy Prime Minister Prawit Wongsuwon confirmed he ordered authorities to investigate into the matter.
"Anyone who have broken the law and caused damage must be punished," he said.
He was referring to a profile published online by the BBC's Thai-language service about the newly crowned king.
The piece, released on December 2 after Crown Prince Maha Vajiralongkorn was proclaimed King Rama X, brought the new king's personal life to light, adding that he did not enjoy the same level of respect as his revered late father.
The article was widely shared on social media and some Thais said its content was insulting to the new king.
Thailand has a strict lese-majeste law against any insult of the monarchy, which carries a penalty of three to 15 years in prison.
A BBC spokesman said the BBC is confident that its article adheres to the BBC's editorial principles to bring impartial, independent and accurate news.
The Thai police arrested a student on Saturday for sharing BBC's article, releasing him on bail the next day.
It was the first arrest under the lese-majeste law since prince Vajiralongkorn became the king.