Is it obligatory to repeat the prescribed prayer that was performed while continuously passing gas due to irritable bowel syndrome, or is it waived because it was performed in that state?

Passing wind nullifies ablution by consensus, whereas burping does not. If you do not find time to pray with proper purification due to passing wind, then your ruling is that of a person afflicted with incontinence: you perform ablution after the time for prayer begins and pray, and what exits does not harm you until the time for prayer expires, and you are not required to repeat the prayer. However, if you prayed while passing wind and you could have found time for purification, then you are required to repeat the prayer.