It has never happened.
Closest was in the 1982-1983 school year:
-- On October 9th, 1982, the Vols football team went to Baton Rouge and tied LSU, 24-24.
-- Three months later, on January 10th, 1983, the Basketvols went to Louisiana and beat the Tigers, 59-58.
Note that the Basketvols didn't even have a single spare point to share with the football team to get them over the hump. Those two games were as close as a football and basketball game could be in combination.
We have beaten LSU in both sports in a single year several times ('01 most recently). Just never both wins in Baton Rouge.