So this may be long-winded, but I gotta get through some theoretical background first!
Conceptual Act Theory
If we take from the Conceptual Act Theory (CAT) of emotions (Barrett, 2014), then we can consider emotions like fear to be what are called "situated conceptualizations." This means that we feel an emotion when we make meaning out of our situated affective experiences (what CAT calls a "conceptual act").
For example, I'm making a conceptual act when I interpret (conceptualize) as fear my high arousal and negative affect in the context of anticipating a snake bite. What this means is that I'm automatically categorizing my experience using an emotion concept (i.e. fear) that I've learned.
As Condon, Wilson-Mendenhall, & Barrett (2014) point out:
Emotion concepts make affective changes in the body meaningful, guide action, allow communication about one’s state to another, and influence another’s mental state and actions.
So categorizing my experience as "fear" is just like categorizing a big gray moving thing with a trunk as an "elephant." The elephant categorization makes meaning of the moving gray mass, allows me to compare it to other objects, communicate what I saw, guides my actions, etc.
Emotion Variability and Pleasant Fear
What's interesting about CAT is that it predicts tremendous variability in emotional experience. The ways in which we can conceptualize our situated affective experiences are vast. In this sense, CAT might predict something like pleasant fear. I've shown you this image before (from Condon et al., 2014), but it's even more relevant here:
As you can see, our prototypical fear concept involves negative feelings. We are very familiar with this negative fear concept, and so we can easily and automatically apply it to our affective experiences.
On the other hand, we never really talk about what it might mean to experience pleasant fear. The pleasant fear concept is not in our folk taxonomy of emotions. We're unfamiliar with it. Thus, we'll have greater difficulties activating that concept and applying it to our experiences.
So, can we learn to experience pleasant fear? Yes!
But at the moment there's little empirical research on learning emotion concepts (in adulthood). We can probably draw from the literature on emotion socialization in children (e.g., Widen & Russell, 2010), but I'm not personally convinced that we can make an easy parallel.
There is potentially some indirect evidence that there is rapid conceptual learning when exposing remote tribes to Western emotion concepts (Sauter, Eisner, Ekman & Scott, 2015, and a response by Gendron, Roberson, & Barrett, 2015).
Also, there's some emerging evidence (Kashdan, Barrett, & McKnight, 2015) that we can improve people's levels of "emotion granularity." High emotion granularity means you label your affective experiences using many different emotion words (happy, sad, surprised, angry, remorseful, contemptuous, etc.). Low granularity means less diverse labeling (happy, sad, good, bad).
So if we can teach people to make more fine-grained conceptualizations of their affective experiences (i.e., improve granularity), you can probably learn to experience pleasant fear.
This is a new and growing area of research, so you'll have to stay tuned for more direct evidence. But overall, you can learn new emotion concepts like pleasant fear, and you can get better at using them with practice.
Emotion Utility (Edit)
It's also possible to treat negative fear as having positive utility (see Maya Tamir's work, e.g., here). That is, even though fear might feel unpleasant, you might learn to view it as serving some adaptive function within certain contexts, e.g., facilitating quick withdrawal from danger.