The S corporation is required to file an income tax return (Form 1120-S) as long as election to be treated as an S corporation (via Form 2553) remains in effect.
See https://www.irs.gov/instructions/i1120s#idm140355079064496
Generally, the president, vice president, treasurer, assistant treasurer, chief accounting officer, or any other designated corporate officer (such as tax officer) is responsible for having the return prepared, signed, and filed.