No, you do not have to pay back the American Opportunity Credit. The American Opportunity Tax Credit is a credit for qualified education expenses paid for an eligible student for the first four years of higher education.You can get a maximum annual credit of $2,500 per eligible student. If the credit brings the amount of tax you owe to zero, you can have 40 percent of any remaining amount of the credit (up to $1,000) refunded to you.
For more information please see the following IRS link: https://www.irs.gov/Individuals/AOTC