Does anyone know which nights are the formal nights on the 10 day Tahitian?
It probably varies by itinerary, as they prefer to have them on days when the ship is at sea, at least in the afternoon, and always avoid having formal nights on days with late ports of call. IIRC, on my cruise to French Polynesia and Samoa, the formal nights fell on Night #3 and Night #8 due to the itinerary.
You should also be aware that the embarkation, disembarkation, and turnaround arrangements on this ship are unusual. Princess has a contract with Omni Airlines to operate a charter flight (777 IIRC) round trip between Los Angeles and Papeete on turn-around day, which accommodates about half of the passengers. The flight leaves Los Angeles in the early afternoon, permitting connections from anywhere in North America, and arrives in Papeete in the early evening. The return flight leaves Papeete late at night and arrives in Los Angeles early the next morning, again permitting "same day" connections to anywhere in North America. Passengers returning to the United States on the Omni Airlines charter flight must go to the airport to check their luggage in the morning of disembarkation day, then return to the ship until final disembarkation between 8:00 and 9:00 PM or so. The bars, the spa, and other facilities aboard ship are open to disembarking passengers on a cash basis on turnaround day. There's also a baggage check for hand luggage aboard ship on turnaround day, at no charge.
Passengers who are not on the Omni Airlines charter flight and who do not have a "cruise plus" (hotel) package also usually have evening departures because all of the commercial flights between Los Angeles and Papeete seem to operate either on a similar schedule to the Omni Airlines charter flight or, alternatively, depart Los Angeles around midnight to arrive in Papeete early in the morning and depart Papeete midmorning to arrive in Los Angeles early in the evening. Passengers on flights other than the Omni Airlines charter flight do not go to the airport on Saturday morning, but rather remain aboard ship until final disembarkation in the evening, and complete check-in procedures in the normal manner upon arrival at the airport. Passengers with independent arrangements in Papeete are free to disembark at any convenient time.
Because of this arrangement, the main dining room serves dinner on an "open seating" basis between 6:00 and 8:00 PM and between 9:00 and 10:30 PM on turn-around day. Passengers on the Omni Airlines charter flight arrive in time to eat dinner during the second window, while those departing on the Omni Airlines charter flight leave the ship late enough to eat dinner during the first window (though those on the first buses to the airport will need to eat promptly when the dining room opens). Passengers who have independent arrangements or "cruise plus" packages before their cruises embark earlier in the day, and may eat during either seating. The assigned seating arrangements for dinner commence on the second night of the cruise.