Does grenadine need to be refrigerated

As a general principle, any ingredients which do not require refrigeration on their own will not require refrigeration when mixed. A notable exception is raw garlic in olive oil; raw garlic carries botulinum spores, which propagate in anaerobic environments. Cold helps to prevent the spores from multiplying and producing toxin, but doesn't prevent it entirely. In addition, you will find texture effects; olive oil is fairly unique among unprocessed (as in, unhydrogenated) vegetable fats in that it solidifies at refrigeration temperatures.

It is likely that prior to use there is minimal likelihood of the ingredients separating as the transport and shelf storage of the item is likely considered controlled. Once in a home kitchen, the problem is not with spoilage (there are plenty of preservatives in it), but that storage fluctuations and use may throw the emulsifiers.

Although I am inclined to dismiss the need for refrigerating it, the is likely that across a few million units, a less stable batch may make its way into a hot and then cold and hot and then cold kitchen and the contents would separate.

For big brands, a major focus is to ensure consistency among products in spite of quality or diverse ingredients; hence watery domestic beer (U.S.), Lay's potato chips, and refrigerated chocolate syrup that is harder to use than when stored at room temp.