It depends on how realistic a game you're running.

In real life, when you pour oil on a sword, the oil basically just runs down the sword, and only a very thin coating will remain on the blade. This coating will either evaporate before you can do anything useful with it, or burn for a moment or two before burning away. In the best case, you get a bunch of burning oil on your hands.

Now, you might be running a more cinematic game, where you want people to be able to pour oil onto their swords and have it be an effective tactic. If that's the case, then you would probably use the same rules that a torch does, and have the sword do 1 point of fire damage in addition to its normal damage. How long this buff will last depends entirely on how cinematic the game is, but it probably shouldn't last more than a few rounds in anything but the most ridiculous of game worlds.

Burning weapons in real life are almost exclusively based on projected fire, or blunt impact of a flaming object. It's not really possible to light a sharp blade on fire without either providing a continual source of flame (like a flamethrower), or using a material that is inherently flammable, which would screw up the weapon's efficacy as a blade.

It's a little bit easier to use a non-bladed weapon as a burning weapon, but it's more likely to damage the weapon that way. The same point about the oil running off applies: even materials that would normally absorb the oil, like wood, will generally be made non-porous when crafted into a weapon. A wooden club that gets moldy when you use it in the rain isn't so useful in the long term. However, if you soak a rag in oil, tie it around the business end of a blunt weapon, and light it on fire, then that weapon can almost certainly be used to cause fire damage in the same way a torch can.

If you soak someone in oil through a thrown flask and then light them on fire, then they will probably take some damage from that fire until they can put it out. If they're wearing flammable clothing, then the fire will likely last until it is actively put out. If they're not, then it will only last a few rounds. The DMG doesn't give a specific number for how much damage fire does, but your instinct of 5 damage per round sounds reasonable.

The rule is an extension of a precedent set by WoTC themselves

The lead rules designer of 5e, Jeremy Crawford, has¹ the power to make official rulings, and frequently does so on Twitter, and in the Sage Advice column on the official D&D site. It's common for him to answer questions with some variation of "if a feature was meant to work that way, it would say so." He has even explicitly stated:

Beware of claims that a rule does something mentioned nowhere in that rule or elsewhere in the core books. There aren't secret rules. (source)

Using this principle, he has made rulings such as:

The Dual Wielder feat doesn't include the benefit of the Two-Weapon Fighting feature. It would say so if it did.
(source)

or

If the grease spell created a flammable substance, the spell would say so. It doesn't say so. (source)

From this, we can derive that barring some explicit clarification from Sage Advice, JC himself, an official errata, or a more specific rule mentioned somewhere else in the game's official material, features in the game are intended to only do what they say. Though, of course, the DM is permitted to make their own rulings and allow spells and effects to do things not directly stated in their description as they see fit.

¹ _{While this status was subsequently backed away from (official rulings only now apply to the Sage Advice Compendium as posted by WoTC, not the tweets from J Crawford), it was true that his ruling was authoritative when this question was asked, and answered; that change in state does not change the overall point.}