I have feeling that I should do it. The difference between open bottle that have stayed outside and fresh bottle is pretty clear. Refrigeration would slow down any reactions and thus keep taste better longer.

Every soy sauce label I remember seeing has said, "Refrigerate after opening." I don't know why this seems to be rarely done (at least in the US).

I keep sweet soy like kecap manis and 醬油膏 in the fridge because occasionally it can catch mold otherwise. I do the same with sweet vinegar like balsamic. However I think this depends a lot on how hot and humid your environment is. In cool and dry climates it's probably not necessary.