Does grenadine need to be refrigerated?
I'm not sure about other types of grenadine, but assuming they're the same as Rose's, they do not need to be refrigerated.

It's not just that they contain sugar - they're practically nothing but sugar, water, and a couple of "chemicals."  There's really nothing in there that can spoil, unless you put something else into the bottle that wasn't there originally.

Unless you accidentally dropped a piece of chicken in there, definitely don't worry about refrigeration.  It can sit on the shelf.