Turkey and Greece are both members of NATO, so if they went to war, would the remaining NATO states have to sit it out? Or would one of the two be designated the aggressor, and the rest of NATO obliged to come to the aid of the other?

NATO commits members to protect each other. If one member invaded another, it seems like a pretty clear violation of the treaty. In other words, grounds for being expelled from pact, so no more conflict about the remaining members coming to the invaded party’s aid.

> if they went to war, would the remaining NATO states have to sit it out

An interesting question, which i think you would need some legal expert on the NATO treaties to explicate. But mercifully for 50 years or so they haven't ever done such a thing, and I don't see the military forces of either country wanting to. The bonkers Turkish leadership is another matter, but the Turkish (and Greek, to be honest) military have a tendency to remove inconvenient leaders.