The rules for Prince Arnaud say:
When you discard Prince Arnaud, choose one player still in the round (including yourself). That player discards his or her hand (do not apply its effect) and draws a new card.
but for the Princess, they say:
If you discard the Princess-no matter how or why-she has tossed your letter into the fire. You are knocked out of the round.
If Prince Arnaud is used on someone holding the Princess, does that person lose due to discarding the Princess, or do they not apply its effect? Given that the wording of Prince Arnaud says "discards his or her hand" rather than "discards the card in his or her hand", does this count as discarding the Princess?