No, you'll just damage or destroy your sword instead.
The rules for Alchemist's Fire say the following:
As an action, you can throw this flask up to 20 feet, shattering it on impact. Make a ranged attack against a creature or object, treating the alchemist's fire as an improvised weapon. On a hit, the target takes 1d4 fire damage at the start of each of its turns. A creature can end this damage by using its action to make a DC 10 Dexterity check to extinguish the flames.
The target is your sword, and it's an object that's within 20 feet of you, so it's a valid target. As a result, when you used the Alchemist's Fire this way, you make a Ranged Attack against an AC of 19 (since your sword is made of steel) - with disadvantage, because it's a Ranged attack against a target at close range.
On a success, your sword will take 1d4 damage per round - and since your sword is a Small, resilient object, it will have 3d6 (10) hit points, so it will take approximately 4 rounds for your sword to be destroyed. However, a creature (including yourself) can take an Action to make DC 10 Dexterity check to extinguish the flames and prevent your sword from being destroyed.
While it's on fire, your sword does no extra damage to enemies, since there's no rules that state that it would.