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Alchemist's Fire never burns out, according to this: How long does Alchemist's Fire burn?

Does that mean I can put it on my sword to make a virtual Flame Blade? According to the rules it does 1d4 damage to an object or creature at the start of each of it's turns, but would that affect a non-flammable piece of metal? I assume that it would add 1d4 fire damage to any damage rolls with a weapon that had the Fire on it.

Being "Sticky and Adhesive" it will not run down onto my hand, right?

So that's 3 questions: will it destroy my sword, will it add damage, and will it hurt me?

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No

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. Adventuring Gear

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 when alchemist's fire is used as an improved weapon and thrown at a creature, that creature takes damage. Its also described as a flask of alchemists 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 50g 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 gold. What you are suggesting is to rather deal 1d4 to as many creatures as many times as you want for 50 gold. 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.

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    \$\begingroup\$ Not to mention the difficulty of safely applying Alchemists Fire to said sword and the possibility of the sword getting extremely hot to hold (though, as you say, this is all stuff the DM would just have to make up if he allows it). \$\endgroup\$ – PJRZ Jun 27 at 13:50
  • \$\begingroup\$ You may also want to mention an existing item that gives a similar-type effect: Sun blade. Radiant rather than fire, but the look is similar. \$\endgroup\$ – NautArch Jun 27 at 14:17
  • \$\begingroup\$ @NautArch are you comparing the 1d8 to undead to the potential 1d4? In my mind you could argue/reason that the qualifier for the damage sets it apart from "always do 1d4" if you wanted to list a precedent for this being unbalanced (50g flask vs rare magical weapon) \$\endgroup\$ – Premier Bromanov Jun 27 at 16:04
  • \$\begingroup\$ Good point, it was more the idea of how the weapon worked/looked. Flame Blade's 3d6 would be compared against the 1d8, so a flame blade is better and both are better than the alchemist fire damage potential. \$\endgroup\$ – NautArch Jun 27 at 16:07
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    \$\begingroup\$ For comparison, basic poison costs 100 gp. It can be applied to one weapon or 3 pieces of ammunition. It forces a DC 10 Con save, with just 1d4 poison damage on a failed save, and only remains potent for up to 1 minute before drying. Obviously, an item half that cost shouldn't do the same amount of damage for an indefinite period of time. \$\endgroup\$ – V2Blast Jun 28 at 5:22
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Aside from the whole point of "the rules say exactly what Alchemist's Fire can do":

Don't mistake the game's rules (or lack thereof) for laws of physics in the D&D universe.

There are a lot of areas where the rules don't get specific about every detail of how something works because it's not, generally speaking, important. The duration of Alchemist's Fire is one of those areas.

For the purpose of using alchemist's fire in a fight, it burns for some indefinite amount of time that exceeds the length of the fight. Even two minutes is usually much longer than any fight lasts, so 'until extinguished' is sufficient definition for the purpose, but that doesn't mean, from a game-world perspective, that it never burns itself out.

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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.

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    \$\begingroup\$ I agree with the ultimate answer, but your entire "targeting the sword" business is all very house-ruley. Where did you get the AC of 19 from? That attack would not be with disadvantage (the rule for ranged attacks is when you are within 5 feet of a hostile creature; your sword does not count as a hostile creature!). Also, objects have a 'hardness' that must be overcome before damage is applied: See the DMG. Unfortunately it doesn't supply hardness ratings for much, so the sword would be a guess, but I would expect it to require a lot more than 1d4 fire damage to destroy a sword that fast. \$\endgroup\$ – PJRZ Jun 27 at 16:20
  • \$\begingroup\$ The ranged attack is specifically for throwing the flask. Surely if you wanted to apply it to an object in your possession, you would just pour it on directly. \$\endgroup\$ – Mark Wells Jun 27 at 20:37
  • \$\begingroup\$ @PJRZ there is nothing house-ruley about this: roll20.net/compendium/dnd5e/Objects The Alchemist's Fire itself explicitly specifies an object is a valid target for the ranged attack roll. This is definitely the right answer for this question. \$\endgroup\$ – Nacht Jun 27 at 22:59
  • \$\begingroup\$ @PJRZ also I believe the hardness you mention comes from either 3.5e or PF. I don't see it in the 5e DMG. \$\endgroup\$ – Nacht Jun 27 at 23:07
  • \$\begingroup\$ @Nacht: Sorry - In 5th edition its changed to Damage Threshold, though there are no examples of it being used (I believe the ships statistics provided in a supplement used it). There is the paragraph on objects and damage types, where it suggests you just need to apply some logic to whether an object can be damaged by something or not. \$\endgroup\$ – PJRZ Jun 28 at 7:10

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