According to contemporary philosophy and cognitive science which mental features resist naturalization the most and why?
By mental features I mean something like qualia, consciousness, mental language, etc.
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John Searle wrote a lot about "problematic features" of mind. His favourites are
You can read about that e.g. in Mind, Brains and Science. I'll try and give a shot summary as well as my personal two cents.
Searle is trying to get rid of the evil dualism introduced by Descartes that has been haunting sciences, not only philosophy, for centuries. It is his agenda to explain how consciousness, as the sum of all mental phenomena, can be explained with what he thinks is "common sense":
The views I have advanced, are, appropriately understood, matters of scientific common sense in that they are, I believe what one would say if one had a modicum of scientific knowledge but was free of the traditional philosophical categories.
Consciousness is a fact, although somewhat mysterious, but there is no reason to believe that science can't explain it. Searle thinks that it's been biologists, mainly, who found an answer to the question what life is, and it will be neuroscientists who will provide an analysis of exactly what functions in the brain are responsible for the phenomenon of consciousness.
The mystery of intentionality is just a point of definition: Searle uses the example of thirst to explain; thirst is a intentional condition, as it always aims at something that quenches it. Thirst is produced by the brain - therefore there is nothing weird about intentional states, and neuroscience will be able to explain it fully, comes the time. Needless to say (pardon my subjectivity) that this is a weak example, and that it is only for his definition of intentionality that allows him to reduce intentional states to a symptom of what's going on inside our body.
Now for subjectivity: As Searle points out, the problem isn't subjectivity, but the focus of the sciences on objectivity. Sciences, he says, collect objective knowledge, therefore they often don't accept subjectivity. But the existence of subjectivity is an objective truth. In his essay Biological Naturalism he introduces the distinction between epistemic and ontological objectivity and subjectivity. Ontologically subjective phenomena can be epistemically objective.
The problem of causality, that has been a major weak point of every dualistic philosophy of mind, can be resolved by accepting a less naive understanding of just what "causality" is. We are not to think of billiard balls, but to compare to chemical structures that have micro as well as macro features. Just as the micro structure of molecules "causes" the substance's macro features, the brain causes as well the micro behaviour, that is, the will or thought of moving, as well as the macro behaviour, which then is the actual movement. It is just two different levels of explanation that have a "common cause", which is the brain. The macro event is realised and caused on the micro level.
I will not go into details with my criticism unasked. If Searle is interesting to you, start with the countless essays. The book I pointed out is not difficult to read at all (in fact it is very vague and naive - oops, I did it again) and can be mastered in a short time.