How should I store homemade salad dressing?
Yes, you should refrigerate. Shelf life of foods which can be stored warm is determined by the fact that they are missing something which bacteria need to survive. For example, flour is missing water. 

When you mix up several ingredients which can be kept in the pantry separately, you never know when you will be adding back into the mixture whatever was missing into the first place. Actually, you have a high chance of doing just that, because bacteria and humans find the same stuff tasty. When you make it more palatable for you, you are probably making it more palatable for them. 

Bottom line: unless you know very well what you are doing, always refrigerate mixtures. They have a shelf life of 3-5 days in the fridge. It doesn't matter what the shelf life of the individual components was.