The two rival Palestinian factions have failed to heal their political disputes since Hamas fighters expelled Fatah from Gaza in a short war in 2007
Explosions could be heard on the airport road and other parts of Tripoli though it was not immediately clear who was fighting whom.
British Airlines cancelled its flight to London, it said on its website. Turkish Airways cancelled its flight to Istanbul.
Social media websites said several rockets had hit the parking area in front of the main terminal.
The airport area has been controlled by militia fighters from Zintan in northwestern Libya since the overthrow of Muammar Gaddafi in 2011 though other groups have challenged them.