It really depends on how quickly they will be eaten.  If you plan to serve them up within 24 hours, it’s ok to store them at room temperature.  But beyond that I highly recommend storing them in the refrigerator.  These brownies have more moisture than the typical brownie which means they can grow mold more quickly.   Vegan date brownies also work great to freeze.  Just ensure that they are in a well-sealed container before placing them in the freezer.  It’s best to eat them within 1 month for the best flavor.