According to an interview with microbiologist for Initial Hygiene Dr. Peter Barratt's in The Daily Mail, the official answer is:"No one kept tomato ketchup in the fridge a decade ago, but then it contained more salt. In recent years, food manufacturers have had to cut the amount of salt, a natural preservative, because of its links to high blood pressure... However, while ketchup could keep safely in the cupboard for a couple of weeks, it's best kept in the fridge and eaten within the stated eight-week period," Barratt said.
The official statement by Heinz is: "Because of its natural acidity, Heinz Ketchup is shelf-stable. However, its stability after opening can be affected by storage conditions. We recommend that this product, like any processed food, be refrigerated after opening. Refrigeration will maintain the best product quality after opening."