Does unearned revenue go on a classified balance sheet??

Yes it goes on the balance sheet.

Unearned Revenue is a liability, most likely a shot term one.

It does go on the balance sheet. And under Classified Balance sheet, it would go under the "Current Liabilities" section.

Unearned income that is income received in advance, is to be shown as a lialbility in the balance sheet.This portion of the income related to the next year for which services is yet to be rendered.