My company pays a small salary to one of the two shareholders of the corporation. This individual resides in Alabama and is otherwise retired. What are our unemployment tax obligations?

Yes. For your state, the earnings are taxed up to the state limit. Let's say your employee made 5,000.00 for the year and the state limit is 8,000.00 so the taxes would be 5,000.00 x your unemployment rate.

If this is a salary, then it is pay, so all taxes need to be withheld, and the company part of the medicare tax and unemployment would all need to be paid.