Does Prince Arnaud cause someone holding the Princess to lose?
The player does lose.  The full rule book is explicit about this:

If you discard the Princess—no matter how or why—she has tossed your letter into the fire. You are immediately knocked out of the round. If the Princess was discarded by a card effect, any remaining effects of that card do not apply (you do not draw a card from the Prince, for example). Effects tied to being knocked out the round still apply (eg. Constable, Jester), however.
When you discard Prince Arnaud, choose one player still in the round (including yourself). That player discards his or her hand (but doesn’t apply its effect, unless it is the Princess, see page 8) and draws a new one.

Using the prince on a player with the princess is one of the main ways to eliminate a player during the round.  The big three ways to eliminate a player mid-round are:

Using a guard (and either guessing correctly or having advance knowledge of their card from the King/Priest)
Using the Baron
Using the prince on a player with the princess