No, it doesn't - but the fire damage doesn't apply the Looting effect. If the drowned dies directly from getting slashed by the sword, Looting is applied. If it's set on fire and the fire finishes him off, the chance of dropping the trident is not increased.
Moreover, if the death from fire occurs more than 5 seconds after the last hit, the chance to drop the trident drops to zero; the drowned will not drop any XP either - the death will be counted as 'natural causes', not attributed to the player.
Above: result of an experiment. 30 cows were killed in each batch, and loot collected:
- First row - a diamond Looting III sword, no Fire Aspect, each cow killed with two hits.
- Second row - a Looting III, Fire aspect II sword - one hit, which the cow survives, then dies from fire.
- Third row - same Looting III, Fire aspect II sword, but the cow is hit twice before allowed to die from fire.
The results are quite clear: the mob killed by fire doesn't drop extra loot.