Do I need to store a mixture of olive oil and balsamic vinegar in the refrigerator?
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.
I don't think so unless it's not pure mixture of oil and vinegar as I think both of them can be stored un-frigerated independently.
Extra-virgin Olive Oil is a product with long life (one year until the next season will produce the next generation of products) and is stored at environment temperature, as long as it is prevented to be exposed to light and heat.
Modena Traditional Balsamic Vinegar is a very long life product, deriving from a long term process during years, environment temperature too.
Mixing them will not change they cold-affinity, so prevent the mixture from direct light and eat, no need for a refrigerator at all.