I have heard that it is possible to lose some rites. I know that you can lose the Rite of Valor on purpose by giving the teeth to Elsa. In fact, not only can you do that, but you should. It's an honorable thing to do (I'm not sure if it actually raises your honor but it should), and if you want to marry Elsa you have to either do this or challenge her in the arena, and doing this is much easier than waiting until day 20: I've gotten so good I always have the game beat long before then if I want.

Are there any other rites you can lose?

I kind of hope you can lose the rite of courage. One time I left Hades without getting the styx water. I restored my game, but I wonder if the game wasn't actually unwinnable after all.

I've played the game a few different ways, but the only rites I've ever lost have been Valor (giving the teeth to Elsa) and Death (I got the styx water, but I'd delayed so long that when I returned I was informed Elsa had already won).

You can also lose the rite of freedom if you take to long Gort will be declared the winner.

1. You can lose the Rite of Freedom by taking too long, even if you capture every village's sigil.

2. I do not know whether you can lose the Rite of Conquest. There is only one General's Shield, so I would guess that you can't, but that's only a guess.

3. You can lose the Rite of Valor, but only by deliberately giving the teeth to Elsa.

4. I do not know whether you can lose the Rite of Destiny, but I would guess that you can by taking too long.

5. You can lose the Rite of Courage by taking too long.

6. I do not know whether you can lose the Rite of Peace, but I would guess that you can't.

7. You can't lose the Rite of Justice.

I guess the next question is, how long is too long?

My question is are these losses automatic if you wait too long? For example, if you wait at the inn for 15 days, does that DQ you and they announce a winner, or do you have to free your particular village anyway to progress?

The Rite of Courage is not automatic; you still need to get some water from the Styx. I don't know about the others.

Makes sense, considering that you absolutely need the key item that you get from there. Nobody else has the opportunity to find it - Elsa and Toro even say something about how Minos searched high and low to get said item, but couldn't.
Even discounting everything else that happens.