Will completing a 3-year ban on IELTS affect my passport? Will there be any record after the 3 year ban?

Yes, the ban may not be noted physically on your passport but could be recorded in digital files associated with it. These records are checked automatically when your passport is scanned at international borders. If the reason for your ban is not relevant to the country you are trying to enter, you are likely to be admitted. However, if it is relevant, you may be denied entry and sent back on the next available flight.