You do not need to refrigerate agave syrup after opening. Storing it at room temperature in your pantry or a cool, dry cupboard is completely fine and generally recommended.
Agave nectar has a very high concentration of natural sugars and a low water content. This creates an environment with low water activity, which naturally inhibits the growth of bacteria, mold, and other microorganisms. Because of this, it behaves very similarly to honey in terms of shelf stability.
In fact, keeping it in the refrigerator can be inconvenient because the cold temperature will cause the syrup to thicken considerably, making it difficult to pour and measure accurately for your baking recipes. As long as the bottle is tightly capped after each use to keep out moisture, it will easily stay fresh in your pantry for one to two years.