Verbal irony is the use of language in a manner that is meant to signify, usually for emphasis, the opposite of what is literally said. Dramatic irony occurs in literature when the true significance of a character's actions, words, and behaviors are fully understood by the reader and yet not fully understood by the character acting/speaking.
In Nothing But the Truth, I would argue that the strongest moment of irony is when we discover, at the very end of the novel, that Philip does not know the words to the National Anthem. This is ironic because the entire conflict of the novel was predicated on Philip's claim that he was punished at school for singing the National Anthem rather than standing quietly. This caused a chain of reactions, including the school's loss of funding, the forced retirement of Mrs. Narwin, and national outrage about how Philip's patriotic values were being oppressed by the school's policies.
I would argue that this is a weird twist on (or inverse of) dramatic irony. The reader knew throughout the novel that Philip was simply being difficult when he chose to hum the National Anthem—he was not expressing any true patriotic sentiments. We were aware that the consequences of this choice were direct result of him lying. However, it was Philip who had the full understanding of the irony of this situation, rather than the reader. The last-minute revelation to the reader that Philip does not even know the words to the song is shocking and dramatically worsens our attitude about his actions. He has effectively destroyed a school district and the career of a beloved teacher, and he has done so all for nothing.