The case has raised questions about whether the Thai justice system favours the rich and famous, allowing them to break the law with impunity.
Vorayuth 'Boss' Yoovidhya missed prosecutor orders to report to court on Thursday, the eighth time he has missed a summons since legal proceedings against him began in 2016.
Mr Vorayuth was charged with speeding, hit-and-run and reckless driving that caused the death of the policemen in 2012.
"He sent a letter requesting to move the appointment but he can't do this again," said Suthi Kittisuppaporn, director general of the attorney general's Department of Southern Bangkok Criminal Litigation.
The Office of the Attorney General said in a televised statement it would seek an arrest warrant for Mr Vorayuth.
"If we wait until the end of the business day and he doesn't show up we will consider this deliberate evasion," Somnuek Siangkong, spokesman for the office, told reporters.
"We will then ask police to seek an arrest warrant from the court."
Attempts by news agency Reuters to reach a lawyer for Mr Vorayuth went unanswered.
Mr Vorayuth is a grandson of the late Chaleo Yoovidhya, creator of the Kratin Daeng, or red bull, energy drink.
Mr Chaleo, 88, was listed as the third richest person in Thailand at the time of his death in 2012, with an estimated net worth of US$5 billion, according to Forbes magazine.
His grandson, Mr Vorayuth, who has spent much of the past five years abroad, including in London and Singapore, according to social media posts, has previously cited work commitments abroad as reasons for not showing up in court.
News agency Reuters was unable to confirm Mr Vorayuth's whereabouts. Some media reported he was in Britain.
In 2012, Mr Vorayuth allegedly crashed his Ferrari into a policeman on a motorcycle in Bangkok's upscale Thonglor neighbourhood.
He is accused of fleeing the scene, dragging the officer's body for several dozen meters with his car as he did so.
Mr Vorayuth's speeding charge expired a year after the accident. The charge of hit-and-run carries a penalty of up to six months in jail and expires in September while the reckless driving charge carries a maximum penalty of 10 years of jail and expires in 10 years, if left unchallenged.
(Reporting by Amy Sawitta Lefevre, Juarawee Kittisilpa and Panarat Thepgumpanat, Writing by Amy Sawitta Lefevre, Editing by Michael Perry and Robert Birsel)