What is the ruling on one who swore a false oath to avert the harm of disclosing a matter, such as passing wind in front of friends?

You must repent to Allah Almighty for the false oath. What you mentioned is not a legitimate excuse for that sin, and you could have used allusive speech (ta'reedh). This oath does not require expiation (kaffarah) according to the majority of scholars; sincere repentance suffices. However, the Shafi'is hold that an oath expiation is obligatory. Therefore, it is more cautious to offer an oath expiation to avoid scholarly disagreement.