So I was at the bank today getting coins. The teller asked me if I wanted a star note. I said yes. So I got a series 1999 $10 note. It looks about VG to me. Is it worth more than face?

Usually they have to be in VF or better unless they are really rare. VG is a fairly low grade for a modern note.

Only way it could possibly be worth more is if it has an interesting serial number (radar, ladder, all the same #, repeater)

In addition, if the note is an error e.g. wet-ink transfer, shifting etc, it will definitely worth more than the face value. Other interesting serial number are low serial number, solid number etc.

Thanks.