Does simple syrup require refrigeration?
The key factor in syrup's shelf life is the water activity in the syrup, rather than the ingredients used to make it. Generally, the water is all 'bound up' with dissolved sugar so microorganisms can't use it to grow, but the lighter the syrup, the more available water it will have. 

In my experience, simple syrup is usually kept refrigerated except for small portions that will be used within a day or two.

For a chart of water activity (aW) of related foods, check here: Water Activity Table