It would heavily depend on what the customer has ordered up to the point of ordering the burger, but in general I'd go with something dark and full flavoured to compliment the taste and texture of the burger. If the customer was a beer drinker, I might offer a Pale Ale, if they were a wine drinker I'd go with an American Red, and if they were more into spirits, I might just suggest a sweet highball, but recommend they have it served tall with easy ice, so there's a constant refreshment while they work through the meal