Of course they do. All humans have emotions. It is how we choose to relate to our emotions that matters. The narcissist tends to repress them so deeply that, for all practical purposes, they play no conscious role in his life and conduct, though they play an extraordinarily large unconscious role in determining both.
The narcissist's positive emotions come bundled with very negative ones. This is the outcome of frustration and the consequent transformations of aggression. This frustration is connected to the Primary Objects of the narcissist's childhood (parents and caregivers).
Instead of being provided with the unconditional love that he craved, the narcissist was subjected to totally unpredictable and inexplicable bouts of temper, rage, searing sentimentality, envy, prodding, infusion of guilt and other unhealthy parental emotions and behaviour patterns.
The narcissist reacted by retreating to his private world, where he is omnipotent and omniscient and, therefore, immune to such vicious vicissitudes. He stashed his vulnerable True Self in a deep mental cellar - and outwardly presented to the world a False Self.
But bundling is far easier than unbundling. The narcissist is unable to evoke positive feelings without provoking negative ones. Gradually, he becomes phobic: afraid to feel anything, lest it be accompanied by fearsome, guilt inducing, anxiety provoking, out of control emotional complements.
He is thus reduced to experiencing dull stirrings in his soul that he identifies to himself and to others as emotions. Even these are felt only in the presence of someone or something capable of providing the narcissist with his badly needed Narcissistic Supply.
Only when the narcissist is in the overvaluation (idealization) phase of his relationships, does he experience the convulsions that he calls "feelings". These are so transient and fake that they are easily replaced by rage, envy and devaluation. The narcissist really recreates the behaviour patterns of his less than ideal Primary Objects.
Deep inside, the narcissist knows that something is amiss. He does not empathise with other people's feelings. Actually, he holds them in contempt and ridicule. He cannot understand how people are so sentimental, so "irrational" (he identifies being rational with being cool headed and cold blooded).
Often the narcissist believes that other people are "faking it", merely aiming to achieve a goal. He is convinced that their "feelings" are grounded in ulterior, non-emotional, motives. He becomes suspicious, embarrassed, feels compelled to avoid emotion-tinged situations, or, worse, experiences surges of almost uncontrollable aggression in the presence of genuinely expressed sentiments. They remind him how imperfect and poorly equipped he is.
The weaker variety of narcissist tries to emulate and simulate "emotions" - or, at least their expression, the external facet (affect). They mimic and replicate the intricate pantomime that they learn to associate with the existence of feelings. But there are no real emotions there, no emotional correlate.
This is empty affect, devoid of emotion. This being so, the narcissist quickly tires of it, becomes impassive and begins to produce inappropriate affect (e.g., he remains indifferent when grief is the normal reaction). The narcissist subjects his feigned emotions to his cognition. He "decides" that it is appropriate to feel so and so. His "emotions" are invariably the result of analysis, goal setting and planning.
He substitutes "remembering" for "sensing". He relegates his bodily sensations, feelings and emotions to a kind of a memory vault. The short and medium-term memory is exclusively used to store his reactions to his (actual and potential) Narcissistic Supply Sources.
He reacts only to such sources. The narcissist finds it hard to remember or recreate what he ostensibly - though ostentatiously - "felt" (even a short while back) towards a Narcissistic Supply Source once it has ceased to be one. In his attempts to recall his feelings, he draws a mental blank.
It is not that narcissists are incapable of expressing what we would tend to classify as "extreme emotional reactions". They mourn and grieve, rage and smile, excessively "love" and "care". But this is precisely what sets them apart: this rapid movement from one emotional extreme to another and the fact that they never occupy the emotional middle ground.
The narcissist is especially "emotional" when weaned off his drug of Narcissistic Supply. Breaking a habit is always difficult - especially one that defines (and generates) oneself. Getting rid of an addiction is doubly taxing. The narcissist misidentifies these crises with an emotional depth and his self-conviction is so immense, that he mostly succeeds to delude his environment, as well. But a narcissistic crisis (losing a Source of Narcissistic Supply, obtaining an alternative one, moving from one Narcissistic Pathological Space to another) - must never be confused with the real thing, which the narcissist never experiences: emotions.
Many narcissists have "emotional resonance tables". They use words as others use algebraic signs: with meticulousness, with caution, with the precision of the artisan. They sculpt in words the fine tuned reverberations of pain and love and fear. It is the mathematics of emotional grammar, the geometry of the syntax of passions. Devoid of all emotions, narcissists closely monitor people's reactions and adjust their verbal choices accordingly, until their vocabulary resembles that of their listeners. This is as close as narcissists get to empathy.