The earliest I've been there was April, which is still not tourist season, but (though it was pretty wet and chilly) you could be outdoors. That's important because Burgundy is a pretty rural place. You'll be done with Beaune in less than a day, so the question is, what else would you like to do there?
The standard prescription for winter traveling is to focus on cities, which have night life and things to do indoors in bad weather. Dijon is close to Beaune, a mere 20 minutes or so, and would make a good base to see Beaune and other sights. Strasbourg has a similar relationship to Colmar; the latter is bigger than Beaune and you might consider Colmar as a base.
Best thing to do is dig into some guide books and see what calls to you. Note that Burgundy is not served well by rail (though the Paris connection is good) so if you want to see more that just Dijon and Beaune you'll want a car.