I recalled that walnut oil was considered a good medium, but said to go rancid easily, so I checked a few sources:
M. Graham & Co. produce a leaflet called Traditional Oil Painting Without Solvents in which they advocate the use of walnut oil and walnut alkyd medium (which they produce) for the superior properties of it as a medium (resistance to yellowing and cracking). It quotes Vasari (famous for for his 16th-century book Lives of the Painters, Sculptors and Architects) saying walnut oil is better than linseed because it yellows less with time. It also says walnut oil "removes color from artists' tools as effectively as odorless paint thinners" and also does not "remove essential oils from artists' brushes".
Mark Gottsegen's Painter's Handbook1 says walnut oil is non-yellowing and dries a little faster than safflower and poppyseed oils. It warns that "like many nut oils" walnut oil doesn't store well and will go rancid unless refrigerated.
Pip Seymour's Artist's Handbook2 says walnut oil was used a lot in the past and, because of its very pale color and brilliant gloss, preferred over linseed oil. It says walnut oil "remains glassy, glossy, and hard-wearing over time, lending colors excellent saturation and depth" and "dries a little faster than poppy oil (3-4 days). It also warns that "freshly cold-pressed walnut oil can quickly turn rancid upon exposure to air".
I imagine how well it keeps will depend very much on how warm a climate you live in, whether it's in direct sun or not, how well the container is sealed, and how refined it is during manufacture. You'll soon notice if it's not lasting well. In fact all oils used for oil painting (including the most commonly used one linseed oil), start turning rancid when exposed to air; it's part of the natural drying process. It's just that the little you've mixed with your paint generally dries before you smell it or you use up the oil before it's a problem.
However, if an oil goes rancid in quantity, such as in a bottle, then you really notice it...! If the oil hasn't thickened too much, you could continue to use it, but you may also find visitors stop dropping into your studio.
When it comes to using walnut cooking oil for painting, a temptation because it's often cheaper, remember that cooking oils often have additive in them which may hamper drying. For instance Vitamin E or other antioxidants are used to increase the shelf life, but do this by retarding oxidation and thus the drying of any paint you've mixed with it.