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.