It seems Unburden doesn't activate with Fling, either that or Pokémon Showdown has a Bug.
I conducted a range of tests using the Pokémon in your Scenario (Landorus-T and Sceptile), I gave Landorus-T 252 EV's in Speed and a Jolly Nature and gave Sceptile 92 EV's in Speed and an Adamant Nature. That was enough for Landorus-T to be faster by 10 in Speed, as Landorus-T had a Speed Stat of 309 and Sceptile a Speed Stat of 299.
In the first test I conducted, on the first turn I got Sceptile to use Fling after Landorus-T used Stealth Rock and on the second and third turn into the battle Landorus-T was still faster.
In the second test I conducted, on the first turn I got Landorus-T to use Knock Off on Sceptile, while Sceptile used Leaf Blade. On the second and third turn in the battle Sceptile went first as it was faster because Unburden activated.
In the third test I wanted to confirm that Fling definitely doesn't activate Unburden, this time Sceptile had a Rare Bone has it's held item, which is 100 Base Power when threw with Fling. After Sceptile threw the Rare Bone on the first turn, Sceptile did not go first on the second or third turn.
Source: As above (credit to my brother for helping me carry out these tests).
EDIT: It appears as though Unburden does actually activate with Fling and it was indeed a Bug on Pokémon Showdown!
(Credit to Malachite who presented me with this information.)
The Bug has been reported on Smogon's Bug Report Page for Pokémon Showdown and should hopefully be fixed sometime in the future.