It is clear from what I can google that rags soaked in linseed oil and similar must be taken special care of because they could catch fire while drying. Is this also the case for rags, etc., with oil colors straight out the tube?
Yes, there is still a risk, as the hazard is in the oil; as the oil dries, it emits heat (an exothermic reaction) which, if it's in a pile of other flammable objects--such as other oily rags, papers, etc--can cause them to combust.
From the Wet Canvas forum (the thread also contains some personal experiences with combusting rags):
Remember it is the linseed oil that heats slightly as it dries (oils dry via a slow combustion process) and if the heat is trapped fire is a possibility, especially combined with highly flammable substances such as solvent. Whether you use solvent or not, soak your oily rags/paper towels in water or spread them out to get air.
And a blog from Mutual of Enumclaw gives an additional citation to rags with oil-based paint on them being a hazard (as well as tips for handling them):
Simply put, rags that contain residue of oil-based paints and stains, paint thinners, varnishes, or polyurethane can spontaneously combust and catch on fire. Here’s what happens: When oily rags begin to dry, they produce heat. Combined with oxygen they turn into combustible cloths that can quickly cause trouble.
(All emphasis above is mine.)
It is clear that the oil is the risk, and as the paints "straight from the tube" contain oil, it is best to treat them as a risk as much as you would the other items.
I think was a problem of long ago. Although it can still happen today, it is now much less common. The risks were more common in the past when someone like a professional painter would collect a large box of oily/solvent/paint soaked rags. Back before water based paints were so common, and before disposable paper towels were available for cleaning, and before various cleaning products were available. A can of solvent type liquid is no problem; the problem is a collection of rags where there is good oxygen access over a large surface area and the center of the pile is thermally insulated by the surface rags.