First of all, I am going to rule out both Pe functions. Their use is merely to take in information from the outside world, and it is objectively orientated, away from the self. This most likely wouldn't cause narcissistic tendencies. Te is also to be ruled out due to it generally not being orientated towards people anyway, although I shall add that it may be used in an unhealthy manner to try to find measurable ways to prove (to themselves or others) that they are more superior. Ni is a contender. It is singular in orientation, towards the user, and often gives the impression that its assumptions are correct. So one I'd say is unhealthy Ni. It could also cause the user to have unrealistic dreams and goals.
There is a difference between narcissism and self-absorption or pure arrogance. Narcissism is an artificial cover to conceal the great negativity they have towards themselves. Narcissism is created out of deep insecurity and fragile self-esteem. It is about themselves in relation to everyone else. They live off other people in an extremely detrimental manner. Thus it makes sense that unhealthy use of Fe would lead to that. Manipulation, seeking constant attention, boastful, hypersensitivity to insults. Ti of course has the duality with Fe, and would fuel the latter function, but the naturally detached way in which it operates means that it probably wouldn't be the primary cause.
About Si and Fi, I am more unsure of. It could be possible that Si gives a similar effect to that of Ni. But Si, with the help of weaker Ne, often takes on a negativistic attitude and sees everything in a pessimistic manner (generalizing here), so perhaps the narcissism wouldn't arise. Fi as a value function could in an unhealthy manner inflate their personal values and beliefs as more important than others. But I think Fi more likely comes under the "self-absorbed" label, rather than narcissistic.
But overall I do not believe it has all that much to do with functions, if at all.