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See Dr. L. Kip Wheeler's definition below:

Situational irony (also called cosmic irony) is a trope in which accidental events occur that seem oddly appropriate, such as the poetic justice of a pickpocket getting his own pocket picked. However, both the victim and the audience are simultaneously aware of the situation in situational irony.

Miss Caroline telling Scout that she cannot read at home but must learn at school—even though Scout is a wonderful reader—is ironic, but not situational. If we refer to Dr. Wheeler's definition, it is apparent that Miss Caroline is not aware of the irony of the situation, and neither is Scout. The only character that would probably be aware of the irony would be Atticus, but he is also extremely diplomatic. Atticus would see the wisdom of Scout continuing to read, as she is learning at a level that is comfortable to her and thriving. However, he also knows that it will only create dissension if he criticizes the teacher that Scout must work with for the rest of the year. For these reasons, Atticus remains relatively impartial in terms of "passing judgment" on this ironic development, coming up with a "secret" compromise.

To figure out what kind of irony is being used, I would refer once again to Dr. Wheeler:

Dramatic irony (the most important type for literature) involves a situation in a narrative in which the reader knows something about present or future circumstances that the character does not know. In that situation, the character acts in a way we recognize to be grossly inappropriate to the actual circumstances...

And eNotes states:

In fictional dramatic irony, the artist causes a character, acting as a mouthpiece, to speak or act in a way which is intentionally contary to the truth...

In this case, dramatic irony would seem to be the more accurate choice. The reader in our novel knows that Scout is very bright, and is encouraged to be so at home. She is clearly more advanced than most of her classmates, both educationally and socially. It is to Scout that the other students look to explain to Miss Caroline why Walter Cunningham cannot, and will not, take the money the teacher tries to provide for him to buy lunch.

So the reader knows something about the circumstances (i.e., Scout's educational level) that Miss Caroline [perhaps] does not know. The reader is very much aware that Miss Caroline is acting in a "grossly inappropriate way" (on more than one occasion) with Scout. While we know that Miss Caroline's reading instruction will be wasted on Scout, Miss Caroline does not—or more likely will not, for some perverse and unknown reason—recognize Scout's level of achievement in reading.

It is for these reasons that I believe the example given reflects dramatic irony rather than situational irony.