I’m seeking an answer regarding the rule of damage caused by igniting oil. When a creature covered in oil takes fire damage, does the additional 5 fire damage apply only once, or does it continue to apply each turn while the oil burns? 

Dousing a Creature or an Object. When you take the Attack action, you can replace one of your attacks with throwing an Oil flask. Target one creature or object within 20 feet of yourself. The target must succeed on a Dexterity saving throw DC 8 plus your Dexterity modifier and Proficiency Bonus) or be covered in oil. If the target takes Fire damage before the oil dries (after 1 minute), the target takes an extra 5 Fire damage from burning oil.

 It is just a one time hit of 5 damage.

Is that text from the 2024 PHB? 

I don't have the book, but it seems similar to the same item in the 2014 version. Is the text you provided the complete text? Because in the 2014 PHB, there are also rules about pouring the oil (flask) on the ground:

Oil (flask)

Oil usually comes in a clay flask that holds 1 pint. As an action, you can splash the oil in this flask onto a creature within 5 feet of you or throw it up to 20 feet, shattering it on impact. Make a ranged attack against a target creature or object, treating the oil as an improvised weapon. On a hit, the target is covered in oil. If the target takes any fire damage before the oil dries (after 1 minute), the target takes an additional 5 fire damage from the burning oil. You can also pour a flask of oil on the ground to cover a 5-foot-square area, provided that the surface is level. If lit, the oil burns for 2 rounds and deals 5 fire damage to any creature that enters the area or ends its turn in the area. A creature can take this damage only once per turn.

Anyway, based on your post, when a creature is covered in oil and takes any fire damage, the oil burns only once and is then consumed.

Thank you for your answer. This is the text of 2024phb

OIL (1 SP)

You can douse a creature, object, or space with Oil or use it as fuel, as detailed below.

Dousing a Creature or an Object. When you take the Attack action, you can replace one of your attacks with throwing an Oil flask. Target one creature or object within 20 feet of yourself. The target must succeed on a Dexterity saving throw DC 8 plus your Dexterity modifier and Proficiency Bonus) or be covered in oil. If the target takes Fire damage before the oil dries (after 1 minute), the target takes an extra 5 Fire damage from burning oil.

Dousing a Space. You can take the Utilize action to pour an Oil flask on level ground to cover a 5-foots square area within 5 feet of yourself. If lit, the oil burns until the end of the turn 2 rounds from when the oil was lit (or 12 seconds) and deals 5 Fire damage to any creature that enters the area or ends its turn there. A creature can take this damage only once per turn.

Fuel. Oil serves as fuel for Lamps and Lanterns.Once lit, a flask of Oil burns for 6 hours in a Lamp or Lantern. That duration doesn't need to be consecutive; you can extinguish the burning Oil (as a Utilize action) and rekindle it again until it has burned for a total of 6 hours.

Oh, now it's better. It clearly differentiates between a creature/object and a space.

Thanks to you!

There was recently about a 20-page thread that discussed and debated the wording of the 2014 version of this item.  The main debate was slightly different than your question from your original post.  Mostly I believe that it was agreed that a single source of fire damage would just cause the 5 additional damage that one time during that one turn.  However, there was a lot of debate about whether or not that consumes the oil or if later on a second source of fire damage would also cause 5 additional damage, and so on until the full 1 minute has elapsed.

Although it looks like the 2024 worded has been reworded and reorganized for clarity, this particular detail seems to remain just as ambiguous as it was in 2014.  It does not say something like "The first time that the target takes Fire damage before the oil dries . . .", and it also does not say something like "Whenever the target takes Fire damage before the oil dries . . .", but instead it just says "If the target takes Fire damage before the oil dries . . ." which can reasonably be interpreted either way.

I'm trying to decide if the removal of the word "any" changes the meaning at all.  In 2014, it says:  "If the target takes any fire damage before the oil dries . . ."

In 2024, it's definitely possible to have the interpretation that the oil is NOT consumed.

It's really hard for me to believe that that's the intent though as that makes this item quite overpowered for a common item that can be purchased anywhere for 1 silver piece.  The 2024 version has even come up with a clever way to reduce the action economy cost for using this item -- changing it from requiring an action to requiring a single attack within an attack action.  So, for this low cost, if the target fails the save you get this effect that's ongoing for 10 rounds that can be triggered multiple times and the target never gets another save attempt or any sort of action to wipe off the oil or anything?  So now, you can have your Wizard, Sorcerer, Arcane Trickster and Eldrich Knight just focus firing the Fire Bolt cantrip onto this creature for several rounds in a row and each of these will now do far more damage than a typical cantrip?

On the other hand, why would someone use one of their attacks to maybe cause 5 damage in the future if certain conditions are met?  That's pretty weak.

So, ultimately the DM is going to have to decide which way to interpret that wording.

Quote from up2ng >>

There was recently about a 20-page thread that discussed and debated the wording of the 2014 version of this item.  The main debate was slightly different than your question from your original post.  Mostly I believe that it was agreed that a single source of fire damage would just cause the 5 additional damage that one time during that one turn.  However, there was a lot of debate about whether or not that consumes the oil or if later on a second source of fire damage would also cause 5 additional damage, and so on until the full 1 minute has elapsed.

Although it looks like the 2024 worded has been reworded and reorganized for clarity, this particular detail seems to remain just as ambiguous as it was in 2014.  It does not say something like "The first time that the target takes Fire damage before the oil dries . . .", and it also does not say something like "Whenever the target takes Fire damage before the oil dries . . .", but instead it just says "If the target takes Fire damage before the oil dries . . ." which can reasonably be interpreted either way.

I'm trying to decide if the removal of the word "any" changes the meaning at all.  In 2014, it says:  "If the target takes any fire damage before the oil dries . . ."

In 2024, it's definitely possible to have the interpretation that the oil is NOT consumed.

It's really hard for me to believe that that's the intent though as that makes this item quite overpowered for a common item that can be purchased anywhere for 1 silver piece.  The 2024 version has even come up with a clever way to reduce the action economy cost for using this item -- changing it from requiring an action to requiring a single attack within an attack action.  So, for this low cost, if the target fails the save you get this effect that's ongoing for 10 rounds that can be triggered multiple times and the target never gets another save attempt or any sort of action to wipe off the oil or anything?  So now, you can have your Wizard, Sorcerer, Arcane Trickster and Eldrich Knight just focus firing the Fire Bolt cantrip onto this creature for several rounds in a row and each of these will now do far more damage than a typical cantrip?

On the other hand, why would someone use one of their attacks to maybe cause 5 damage in the future if certain conditions are met?  That's pretty weak.

So, ultimately the DM is going to have to decide which way to interpret that wording.

 You make a really good point. Looking at the 2024 version, you can use the "specific beats general" method, and see that when the oil is used in Dousing a Space, it specifically says that the oil burns for 2 rounds (12 seconds) and that the creature only takes the extra damage once per turn. The Dousing a Creature or Object portion does specify length of affect (1 minute), and doesn't say anything else regarding limiting the damage to once per turn or limiting the timing due to "burning". It does say "If the target takes fire damage before the oil dries (1 minute), the target takes and extra 5 fire damage from burning oil." 

Having the fire damage last for the whole minute does make sense to a certain extent. When dousing an area, it will spread the oil far thinner when over a 5ft square than over a creature (although creatures larger than Medium start to make it make less sense). It makes sense if the oil is more concentrated that it might burn longer (1 minute vs 12 seconds). However logically it wouldn't make sense that it would need to be reignited by fire damage each time, instead of just giving a burning type condition. 

They definitely could have worded this better. As written, unless someone sees something I don't that makes the rule explicit, I wouldn't have a problem with a player arguing that the extra fire damage would occur whenever the target took fire damage for 1 minute if doused.

Much of this is a leftover from earlier editions where the oil did somewhat more damage and burned longer making it more useful for dealing with fire vulnerable creatures like trolls and white dragons.

Quote from Wi1dBi11 >>

Much of this is a leftover from earlier editions where the oil did somewhat more damage and burned longer making it more useful for dealing with fire vulnerable creatures like trolls and white dragons.

 I think prior trolls needed consistent fire damage to stop regenerating.