Am I required to amend my tax return if I overpaid my taxes?

People make honest mistakes in filing their returns. And the IRS makes honest mistakes in entering data which makes it look like you owe more (or less) than you do.

I have received several notices from the IRS over the decades noting discrepancies between what I reported and what they received. They just politely asked for more money. Usually they were right, but in one notable case, they entered several items of data incorrectly and presented me with a notice saying I owed them several thousands of dollars more. (This was shortly after a government shutdown, when they had large backlogs and were overworked.) I examined their letter and my return and wrote them back refuting every point they made (politely, of course). They wrote me back, not exactly saying that I was right, but saying that, upon reviewing the correspondence, I owed them $0.00.

The IRS is not going to audit you solely for a discrepancy of $13.00. If the discrepancy was such that you owe them money, the IRS will send you a notice to that effect. In your case, I don't know if they will just send you a check for $13.00, but they might.

This is an interesting question because the IRS rarely tells you that you don’t have to do something. Typically they just list things you’re required to do. For this situation those are filing a return and paying taxes on time, which you appear to have done. I can’t find anywhere a requirement to amend a return where you are owed money. That seems to mean you don’t have to, but I can’t find a way to confirm that in any way you can rely on.

I can say I wouldn’t bother for that amount. In one case that I didn’t catch, the IRS did send me the money I was owed without me doing anything.