It means that they don't enjoy even the tiniest little bit of it, or, in other words, they don't enjoy it at all. They don't enjoy the gold, they don't enjoy the jewels, they don't even enjoy the smallest brass ring... All they do is guard it. A brass ring is the least valuable thing that you might find in a hoard of treasure. It's the equivalent of saying that someone has millions of dollars but doesn't enjoy a cent of it.

When I read it, I assumed that it meant dragons can't enjoy gold. They are dragons. They have no idea what it's worth, what it can do. They might gather lots of it, but they can't spend it, so they can't enjoy it.

As far as I can tell, there are two possibilities here. 1. "A/the brass ring" can be used to mean "the best or most precious prize"; if we apply it to this context, Tolkien's statement means something like, "...and they guard their plunder as long as they live, and never enjoy the best part of it." 2. A second possibility comes to my mind after recalling a the children's lullaby Hush, Little Baby (this is actually the only instance that I can recall of hearing "brass" used to describe a ring). One line of the song reads: "If that diamond ring turns brass, Mommy's gonna buy you a looking glass." This suggests that a ring being brass is an undesirable quality. If we apply this to Tolkien's statement, then it means something like: "...and they guard their plunder as long as they live, and never enjoy even one tarnished old ring." It seems to me that the second possibility is most likely, given the context. Certainly, I'm no expert on Tolkien or mid-20th century English, so others may have better interpretations of this.