If you aren't required to file a return, there may not be any reason for you to use Form 8822. But, if you usually file tax returns and you're planning to file your next one soon, you can simply enter your new mailing address directly on the tax return form instead of using Form 8822. You also have the option of writing a letter to the IRS that includes the same information that Form 8822 requires.
Failing to update your mailing address with the IRS may have some consequences if time-sensitive materials are sent to your last known address. Under the law, you're deemed to have received anything the IRS sends to your last known address – meaning the agency doesn't have to give you additional time to respond to a letter demanding that you pay additional tax, for example, just because you forgot to update your mailing address after you moved.