With virginias, I don't like anything that has too much of a strong flavor. Iced tea is always great (no sugar for me) with Virginia.
Burley blends that are more on the dry side (HH Burley Flake, for example) I think go great with a little sweetness, so bourbon is a great match, or a Manhattan. I also enjoy Guinness with burley blends that are already a little sweet, especially with a little cocoa flavor (H&H Classic Burley Kake, MacBaren London Blend).
I paired some sipping rum (it was Mount Gay XO, and no, I didn't read that as an imperative (not that there's anything wrong with that)) with both C&D Blockade Runner and Redburn, and that was very nice.
But beer always works for me with anything