Don't know whether it's just my appreciation, but almost ALL foods I've bought in the States say "refrigerate after opening". Perhaps it's just the FDA being ultra careful.I’ve NEVER kept my Furikake in the refrigerator!!
I don't see any reason at all to put something that has sugar AND salt in the fridge.
I don't put jam, chutney or hot sauces in the fridge, nor pickles, gherkins or eggs, and the only reason butter gets refrigerated is that it would melt in the tropical heat outside.
As for the warning; I get the same thing on mustard oil. "For external use only", when it's actually the most divine thing with potato and spinach curry!