Is it possible to permanently lose the contents of a treasure box or can you somehow return and redo something to respawn the box?
Treasure boxes have, and probably always will be, one time occurrences in the Final Fantasy series. Therefore, if you sell a unique item you will permanently no longer have access to the item (I found out the hard way with the bloody
Morph materia in FF7). There are no missable items in FF13-2 so unless you sell it, you can get it.
The only exceptions to these are mini-games. Although I don't think any mini-game in FF13-2 gives a treasure box as a reward.
Another point to note is the
Live Trigger rewards. You get rewards, kind of like the treasure box, for a 'funny' answer and a 'serious' answer. You can use
Time Reversal to replay these events and retrieve the missed items. But once you have both items, they cannot be gotten again.
Only if you sell the contents ;) otherwise you'll be able to obtain them in any time line, unless it's already opened, be it in the future or past.
There is also no way to respawn the treasure balls, time reversal doesn't work either (the same thing goes for treasure cubes).
Hope this helps you out.