ah, yes...the joy of OSRIC...all the 'flavor' of 1st edition...the tradeoff between rules clarity and perceived freedom.
Looking at my copy, I can find no instances of the word 'splash', no mention of oil in the missile weapons table, etc. Wondering at this point if it's my observation/search skills, or more 'flavor'.
In the absence of real data, I would be tempted to use (NOT THE OFFICIAL 1E answer but) my own house rule from WAY back and say that a flask of oil on direct hit will do 1d6 damage the first round, then if nothing is done the second round then it will do a second 1d6. If we're really talking lamp oil and not some alchemical construction, I would be tempted to ignore any possibility of splash damage, and for balance reasons I'd avoid any temptation to have 'save or catch fire' rules...though I might make items potentially damaged (scrolls, etc.) in cases of high damage. If someone spends their turn putting themselves out give them no second 1d6 or an additional 1 point.
The good and bad of Osric like its source is that you are allowed/expected to rule on a lot of stuff yourself...though if I really haven't missed anything and rules for oil are not in there, I would consider it an error to have formalized it as a proficiency without the detailed rules.
Also note that I consider this damage from lamp oil to be a bit too high for a 'real world' view, but better from a game drama perspective...I don't think a good S&S epic benefits from having oil do really no damage except for a singed sleeve, unless you roll a bad save and then you are completely engulfed because you failed to stop, drop, and roll.