Police were considering charging Sheikh Khalid Hamad Al-Thani with reckless driving and other offences before he - and his cars - disappeared.
The Thani family rule Qatar, but it is unclear whether Sheikh Al-Thani is part of the Arab state's royal family.
A video shows his yellow Ferrari racing a white Porsche down residential streets, passing a stop sign.
The video ends with the Ferrari's engine smoking as the sports car pulls into a driveway in the affluent section of Los Angeles.
Sheikh Al-Thani has been dubbed the "patron sheikh" of drag racing, according to the Los Angeles Times.
He has given millions to Al-Anabi Racing, and has been photographed driving on racetracks a number of times.
Police said the cars in the video were driving recklessly.
However, officers who arrived to find the cars in the drive were unable to make an arrest or issue a citation as they hadn't witnessed the incident.
Investigators have connected Al-Thani to the car, but they are yet to confirm whether he was driving either vehicle as the drivers' faces weren't clear in footage or pictures.
The footage drew further attention after police said someone involved claimed diplomatic immunity.
Lieutenant Lincoln Hoshino, a spokesman for the Beverly Hills police, said the car was not registered with the State Department to be brought into the US.
Mr Hoshino also stated that it was unlikely the man had diplomatic immunity.
"It's against a federal law for someone to claim diplomatic immunity when they don't have it, so we're looking at that and then we're also looking at the reckless driving," Beverly Hills Police Chief Dominick Rivetti told reporters.
Beverly Hills authorities said that they may look to prosecute those involved, and that the Qatari consulate has been contacted.
The video has since gone viral and has now been viewed over 1.2 million times.
A Qatari sheikh has abruptly left the US after his yellow Ferrari was caught on camera racing through Beverly Hills.