No, alchemist’s fire doesn’t work this way
Assuming that it does burn eternally, by RAW it can’t make your sword into a Flame Blade. Alchemist’s fire is very specific about how it is to be used. The description of alchemist’s fire states:
This sticky, adhesive fluid ignites when exposed to air. 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
This doesn’t mean that whenever alchemist’s fire makes contact with a creature, they take 1d4 damage. It means that when alchemist’s fire is used as an improvised weapon and thrown at a creature, that creature takes damage.
It’s also described as a flask of alchemist’s fire. A flask’s worth of alchemist’s fire will deal 1d4 damage.
So if you need some logical reasoning, coating your sword and hitting a creature would be a lot different than throwing a flask of napalm at a creature. Water balloon versus wet foam sword.
As always, your DM can rule differently.
I think we needn’t dive into the balance implications of this effect. You’d be spending 50 gp to add 1d4 fire damage on hit (that needs to be extinguished) to any weapon forever (or the duration of the fight). Consider that the flask as intended is meant to deal 1d4 damage to one creature every turn until extinguished for 50 gp. What you are suggesting is to instead deal 1d4 to as many creatures as many times as you want for 50 gp. Well, I suppose we dove anyway. But you can see why your DM might (should) frown upon this request.
As far as destroying the sword or dealing damage to you, that’s up to your DM, should they allow this maneuver. There aren’t any rules listed for the item regarding applying it to a sword, but we might infer that swinging around a metal stick covered in napalm could prove… unwise.