I seem to remember reading to make white wine look red it's two drops red food colouring and one drop blue. I'd just have it in the bottom of a glass, and cradle the glass like a snifter to hide it. Pour the white wine in while talking, look down to see it red. The misdirection of the joke and hand holding the glass (admittedly in the wrong manner for white wine) will cover the change. Simple, cheap, and easy for the non-magician. Job done!

Indeed, a 2:1 ratio of red to blue will do the trick in giving it that burgundy-esque look. The change is well hidden if you are using the aforementioned mirror glass.