Refrigerating sauce aft. opening

Probably not. I stress probably. Smell it first and if it's nice, than give a small taste test. If you get a gagging sensation, then toss it. Your body is more sensitive than your taste buds, so trust it to tell you... but if you have doubts, toss it.

Traditional teriyaki sauce, if made with traditional fermented - ie fermented for at least a year prior to bottling and not pasteurized afterwards - soy sauce, and sake and other traditional ingredients will keep at room temperature a good long while, sometimes months, sometimes longer, and it will only improve the flavour over time.

HOWEVER, most store bought sauces have been pasteurized, or chemically 'fermented' (ie, the flavour is made with a purely short term, chemical reaction, not with bacteria, mould or yeast as is traditional) which does not have the good mould and yeast in it to keep the sauce fresh at room temp. So many commercial sauces need more care once opened to keep them safe.

FYI, Soy sauces is made with koji mould, so is sake, the mould breaks down the starches in the rice and beans to feed the yeast and bacteria - all of which are good for you in moderation. If mould is freaking you out, think how yummy blue cheese is.