He collapsed during the second half of Saturday's game against West Bridgford at the Rutland Showgrounds.
"The club is grateful to the ambulance service who were promptly in attendance," an Oakham statement said.
"Despite their best efforts, Savanaca was pronounced dead on arrival at Peterborough Hospital."
The statement added: "The thoughts and prayers of the club, its players and supporters are all with Savanaca's family at this very difficult time."
Oakham play in Midlands 2 East (South), the seventh tier of the English rugby union pyramid.
Rugby union player Savanaca Kokoibulileka has died after collapsing on the pitch while playing for Rutland-based club Oakham RFC.