Sunflower oil is the oil that results from pressing the kernels of sunflower seeds. Although seed oils, as a general rule, are not very Paleo-friendly, there is more to the story.
Paleo and PUFAs
Most of the reason we avoid seed oils is because they tend to be very high in polyunsaturated fats (PUFAs), particularly the pro-inflammatory omega-6 fats. The increase of seed and vegetable oils in our diets lately is a large part of the reason we have seen a rise in obesity and chronic illnesses, such as diabetes and heart disease. To avoid all these nasty things, we’re better off limiting our omega-6 intake and increasing our intake of omega-3 fats so they are at least at a 1:1 ratio (ratios vary, depending on who you ask, but you never want more omega-6s than omega-3s).
Polyunsaturated fats are also very unstable when exposed to heat, light, and air, all of which they tend to get exposed to, either in the factory, or on store shelves. When this happens, the fats oxidize and oxidized fats lead to oxidized cholesterol in our blood stream. This is very dangerous because it is the oxidized cholesterol that will latch onto our arterial walls, creating plaque and inflammation.
Linoleic Acid v. Oleic Acid
However, sunflower oil comes in a few different varieties. High-linoleic sunflower oil should indeed be avoided, but high-oleic sunflower oil consists mostly of oleic acid, a monounsaturated fat that is also found in olive oil and lard. Monounsaturated fats are much more stable than polyunsaturated fats, so you don’t have to worry about them going rancid on the shelves. If you can get your hands on the good, high-oleic acid oil, feel free to go for it, but you’re still better off going with butter or coconut oil as a general rule.
Mark Sisson gives his breakdown of sunflower oil in this post.