The oily thing is strange. One would think that if an oily liquid was coming through the wall that the plaster or drywall would somewhat filter out the oil and just let the mositure through.
I think if it was venting from a stove that it would have a blackish color to it.
I wonder if you rub it with a damp cloth what would happen. If it foams up then it would probably be soapy condensation from the bathroom
As you can tell, I'm doing some guessing here.

Post is confusing. "an odd wet spot on a wall", "the drywall is not damp and seems hard". Is it wet or not? Is it drywall or plaster? If you're speaking about a kitchen stove, they don't vent anywhere unless it's a downdraft unit.
Some questions. How recently did the spot appear? How big? Did the spot appear after a shower, bathroom use or rain? Does it get bigger or wetter with the use of any of the water sources nearby? Have you tested each water producing fixture to eliminate them? How far off the floor is this mark? Does the spot have marks running down the wall? Is it possible that someone spilled something on the wall?
If it has stayed wet for more then a few days and is not drying out, open the wall. If it's a plumbing fixture or pipe, it will be right behind the wet spot.
Ron

Thanks! Ron, good questions.

We just bought the house and have been there 3 wks and it just appeared but my husband believes it was painted over (pre existing). It's plaster actually, and it is wet, can wipe with my hand, but the plaster isn't bubbly (my old place had drywall that did that so maybe I just expected that with wetness). It's about 2 ft across adn 2ft high, about 7 ft up on the wall, a ft or so from the ceiling.
. No, nothing was tossed up there or spilled. :) It's been wet at least 2 days. I"ll have to measure it and watch how it changes as we use things. I am going to wipe and dry itoff and see if it reappears quickly or takes awhile. Then we may have to open the wall. Ugh, fear as a new home owner that this is the beginning of many surprises.

Does it line up with the shower head in the bathroom? If you put your ear to the wall do you hear water running if you turn on the nearby faucets. You would do this one faucet handle at a time. What are the pipes made out of? Copper, galvanized, pex, etc...? How old is the house? Possible pin hole leak in a supply pipe. Possible hole in a drain pipe. If it's a supply pipe, it will get wetter and wetter and run into the basement or on the floor. The drain will only leak when that fixture is used. The spot will stay pretty localized unless the offending fixture is used alot on the hole gets bigger.
Still think you need to open the wall. Plumbing doesn't heal.
Ron

The wet plaster probably won't bubble like your drywall did when wet. It does sound like something is leaking. I would suggest that you check everything in the bathroom to see if there are any visible leaks before you open up any walls. Is there any dampness around the toilet...under the lav? Is there an access panel behind the shower/tub that you can check with a flashlight? Hopefully if it's a leak, it would be out where you could get to it.
Ron is right, if you can't find an exterior problem, you will be forced to open something up to find what's leaking.

PLaster can wick and hold a pretty decent amount of moisture.
The problem usually occurs as it dries out again and the bond between the finish and scratch coats fails.
Repeated cycles can cause the plster to change form (chemically) and occupy more volume.
The finish coat than starts to crack and come off in pieces.

I had a similar mystery spot once. Turned out that a vent that ran up that wall to the roof had had the caulking (on the roof) chewed up by a critter and rain was leaking, dribbling down the pipe to where it touched the wall; resulting in wet spot on wall nowhere near water pipes. Look for a pipe sticking out of the roof in line with that spot. Or open up the wet area to look for the source. It is not that hard to re-plaster a small hole.

raee, excellent point! Old flashing around the vent pipe is a prime candidate for a leak, especially if it's cast iron and has been tarred at one point. Hub is close to or touching the wall and water just runs down the pipe. If the spot occurred at about the time of a rain or a little after, that might be the issue. Inspect the pipe flashing and seal/replace if necessary.
Had that happen at my house. Noticed water in the hub in the attic and water stains going down the pipe. Once I sealed the stack flashing the issue was gone.
Ron

Actually I have a similar situation new dry wall sudden oil spot on ceiling and what looks like oil drips on several walls like it seeped out of the drywall through the paint. Really weird.

This is happening to me too. They are tiny little wet spots, like the size of a pin head. They feel a little oily.

I’m seeming oil dripping on the wall behind the light fixtures what is causing it ?