Technically speaking, you don’t have to store opened bottles of ketchup and mustard in the refrigerator.
But it’s a good idea all the same, since you’ll get a lot more mileage out of them if you do.
Due to their high acid content, both mustard and ketchup will keep fine for about a month in the pantry. But after that, their flavor, appearance and texture will begin to break down and deteriorate much more quickly at normal room temperatures than in the cooler 40° F storage conditions of the refrigerator.
An opened bottle of ketchup should stay at peak quality for about 6 months in the fridge mustard for at least a year. And both will remain safe to consume for much longer than that, as long as they’ve been continuously refrigerated.