I’m writing a novel and one of my characters speaks in poetic verse, should I format her dialogue differently to my characters who communicate in the traditional manner?
What type of effect are you looking for with the story?
If you're looking for a realistic impression, then I think different formatting would risk breaking the suspension of disbelief. Doing italics or something like that could easily distract the reader.
I also think different formatting could interfere with showing the oracle speaking.
Think of it like this; If it was a movie instead of a book and all actors behaved like movie actors, but the oracle acted as if on a theater stage instead. (Articulate, speak too loud, use too large gestures...) It would look unnatural... (I've seen it when nationally acclaimed stage actors try in front of the camera and fail miserably...)
Here's an example of how italics could ruin a text when used for words from other languages. And if you don't want to watch the movie, the gist is this: you italicize words when you want to put emphasis on them, like in any other situation (or in this case, language).
I think the formatting would cause problems like the ones above.
I am assuming the oracle speaks in poetic verse either because that's how the powers they get their prophesies from speak, or because that's the only way they can connect to their prophetic vision.
It would still come out of a mouth just like for every other character and if you saw it, it would be a person speaking, perhaps (but not necessarily) in a cadence and with some odd word order, but probably not a dramatic stance or all that. You might not even catch on to the poetic verse at first. (An effect you can give the reader by not formatting it differently).
But what if the oracle does in fact act as if on a theater stage? You show that with action and description. Not formatting of the text.
If, on the other hand, you want to do different formatting as an artistic/abstract statement, and perhaps are doing it with other parts of the text as well, then you could use everything from italics, to actually formatting the text as a full poem... If you can convince the reader this isn't just a text, but that something else is going on as well, they'd accept that...
I don't see a problem in differentiating one character's dialogue.
However, if you do decide to differentiate the dialogue, you must first make sure that the one character speaking poetry, will speak poetry 100% of the time.
If that is true, you can italicize their dialogue to differentiate it from the regular dialogue.
But, if they speak poetry sometimes then this won't work. It would be too confusing to the reader if you switch between italicizing and non-italicized for one character's dialogue.