The issue of refrigerating condiments always sparks a lively debate. Some people can’t stand cold ketchup; others won’t touch it if it’s been out on a table for a week.
The official answer on mustard, from the French’s website, is that mustard doesn’t have to be refrigerated for safety, but it does for quality. Mustards keep their pungent flavor and their color longer if they’re refrigerated.
While the acidity in mustard will slow down bacteria growth, it may not stop it completely. So if you have a jar that you stick a knife or spoon into, you might want to be more careful with it than with a squeeze bottle that’s less likely to be cross-contaminated.
Ketchup also will keep longer refrigerated, but it can keep for a few weeks at room temperature as long as it’s in a cool, dry place and is out of direct sunlight.
— Kathleen Purvis