I don't think that I have the sufficient information to type these people as for the most part I have seen them act in school or in battle where they had to conform to a certain code of a social ethic.
I would have to guess that Harry Potter is an Introvert. When he had free-time in school, an environment where he had many friends and an opportunity to socialize, he didn't bother. The environment encouraged him to socialize, yet somehow he consistently displayed a disinclination to do so.
I would guess that he also is intuitive as in friendly and laid back conversations with professors, he displayed an ability to connect ideas together well and also enthusiasm for doing so. Seems to me that this behavior came to him rather naturally.
Would also say Thinking type as in almost all situations where he was able to behave naturally, he sought for structure in notions as well solutions to various problems. Rather rarely he focused on personal sentiments or aesthetical impressions. Judging or perceiving, I would probably guess Judging, but this is the weakest of all inferences I am making. I am drawing this conclusion on the basis of how Intuition seemed his dominant function as he did not seem to be very fond of structured thought. Nor did he seem to be a T dominant, as he did not have this natural intense focus on the impersonal and the characteristic dryness we associate with T doms especially Ti doms. In that regard he contrasts with another character in the plot; Snape, whose normal mode of utterance was very sequential, carefully pondered, dry and as precise as possible. (One may argue that this is attributed to the fact that he is old and a professor rather than to his type, but I'd be inclined to think differently: other old professors did not have those characteristics. Its also the case that most of Snape's personal experiences weren't different enough from theirs to regarded as the sole cause of the significant differences in character between him and them.)
Hermione, its also a tough decision. I would have to say Introverted as she also made little effort to interact in an environment where she was encouraged to do so. Would say Intuitive as she had a seemingly natural fondness for abstraction. It doesn't seem to be the case that she was just storing data in her brain like an encyclopedia. She managed to apply her knowledge to various challenges in order to solve the problem. Its one thing to just remember a notion or to have a vivid impression of it which could be accomplished primarily by sensing, but it is a whole another matter to connect your impressions to various problems that you aspire to solve. Doing that requires connecting ideas together, brainstorming or in other words using abstract reasoning which in almost all cases is resultant of the employment of Intuitive faculties.
An interesting observation about this phenomenon can be made in a study of mathematicians. You'd see that at the beginning levels people who aren't talented at abstract reasoning will manage to memorize enough proofs to get through their exams, but once they get to theoretical mathematics, this no longer becomes doable and they begin to struggle due to their inability to think abstractly.
I would also guess that Hermione is a Feeling type as she had a rather intense focus on her immediate social surrounding, such as Harry and Ron. It doesn't seem to be the case that she was un-naturally motivated to act in this manner. She was dealing with people she thought she could trust.
I would also say Judging, as Intuition seemed to be her dominant function. I would guess so because she very easily chose to pursue the pursuit of abstractions instead of building relationships with people or satisfying aesthetical tastes when she was encouraged to do both. She had close friends and professors were friendly to her and there were paintings and spells that have been of aesthetical interest, yet chose not to indulge in that a great deal. She was also encouraged to study and cultivate her talents. Seems to me that there was no extrinsic pressure for her to choose either option.
Regarding the J faculty of both characters, it seems to me that it was also their nature to go about the external world in a sequential manner rather than chaotic. Children who have a perceiving preference tend to get overwhelmed with perceptions coming from the outside and often struggle to organize them. This does not seem to be a part of the natural dispositions of either Hermione or Harry.
The above thought-experiment was conducted under the assumption that there is a world somewhere (even if purely conceptual and imaginary) where Harry Potter and Hermione exist. Hence, in that event their natural dispositions would be described as above.