Most hot sauces are pretty inhospitable to foodborne illnesses, and can safely stay in the cupboard rather than the refrigerator. Nevertheless, the flavors in the sauces will break down over time, storing them in the fridge as opposed to room temperature will slow deterioration and keep the sauces fresher longer. If you use your sauces quickly enough then you probably won't notice a difference, but if you like to keep them around for awhile it's worth the fridge space.
There is an argument that making hot sauce cold will dull the flavor, and this is true. However, given how small the amounts used are they will come up to temperature very quickly, a dash of cold sauce on hot food will be warm by the time you get it to your mouth, so it's not a good reason to keep hot sauces in the cupboard.
The only time I would store a hot sauce at room temperature is if it will thicken up too much to come out of the bottle easily (presuming it does not say it requires refrigeration of course). Then I will keep it in the cupboard (not exposed to sunlight).