It follows the lifting of some sanctions against Iran in January and the re-opening of the British embassy in the Iranian capital in 2015.
BA said it will run flights six days a week from London Heathrow.
The Boeing 777 will depart just after 21:00 BST on Thursday.
BA, which offered the first scheduled flights between London and Tehran in 1946, ended its three-times-a-week service in October 2012, a year after the British embassy was closed in the Iranian capital.
The embassy had been stormed by protesters during a demonstration against sanctions over its nuclear programme.
BBC diplomatic correspondent James Landale said BA’s decision to resume direct flights may help improve the UK’s relationship with Iran.