This answer argues that, so far, no story has really "featured" a full orbital ring and only one or two have featured partial orbital rings. To appreciate this perspective, we need to first need to establish a working definition of an orbital ring.
Paul Birch and Anatoli Yunitski both wrote articles about orbital rings at around the same time in 1982. (I don't think that Nicola Tesla, while recovering from Malaria, was really onto this idea.) Birch's ring was american-football-shaped with two space elevators that connected to the pointy ends.
Birch's paper was supported by good math; therefore, I think this is the best original definition of what I would call a non-fictional and perhaps technically viable orbital ring. It is an inertially supported active structure, or rather, a megastructure.
Yunitski's design was circular and published in a soviet youth magazine. I believe that, even to this day, its proposed operation relies on some flawed momentum transfer math, but it is still fine as a sci-fi concept and he still appears to be promoting it.
Paul Birch also talked about "Partial Orbital Rings Systems" in his article. Keith Lofstrom published on the Launch Loop around the same time, and the Launch Loop fits within Birch's broad mathematical definition of a Partial Orbital Ring System. However, the Launch Loop (aka Lofstrom Loop) is a more refined and concrete engineering proposal.
My answer will, therefore, cover both full and partial orbital rings as defined by Paul Birch, and include the Lofstrom Loop, which was proposed by Keith Lofstrom. It will not include ring-shaped orbiting structures if they do not use active support or if they appear to just be structures that are supported by, or make use of, an unspecified form sci-fi magic, such as artificial gravity, impossibly strong materials, force fields, etc.
With these constraints in place, there are limited references.
Fred Pohl used launch loop in two of his Heetchee novels (note: this is second hand information as I haven't personally read these novels).
Arthur C Clark and Frederik Pohl wrote a novel called "The Last Theorem" in which a student askes a professor about the Lofstrom Loop...
Even Artsutanov’s lift wasn’t the only game in town. What (asked one
of the students one day in astronomy class) about something like the
Lofstrom loop? For that you didn’t have to start by putting some
humongous satellite into orbit because the thing just sat on Earth’s
surface, from which it flung your space capsules into orbit. But there
Dr. Vorhulst began to rein in the class’s speculation. “Friction,” he
said succinctly. “Don’t forget friction. Remember what reentry did to
a lot of the early spacecraft. If you used a Lofstrom loop, you’d need
to accelerate your capsule to that seven miles a second of escape
velocity that I was talking about the other day before you let go of
it, and then the air friction would burn it right up.”
Of course, the part about "air friction would burn it right up" is nonsense but at least the technology got a nod from two of science fiction's great authors.
Robert G. Williscroft featured the Lofstrom Loop in his 2015 novel entitled "Slingshot". I'm thinking that this might actually be the first story to feature an orbital ring - albeit a partial one.
The Wikipedia article on Orbital Rings provides several examples of orbital rings in fiction that I disagree with (at the time I wrote this, that is). For example, the ring-shaped settlement in Neal Stephenson's 2015 book Seveneves. This is an orbiting ring-shaped settlement, but not an "orbital ring" by the definition I provided above. Similarly, while Arthur C Clark's 1979 novel "Fountains of Paradise" features a space elevator, it predated the 1982 publications on Orbital Rings and did not really feature them either. Rather, on page 254 a character name Morgan contemplates the idea of connecting geosynchronous satellites together to form a ring.
Later, in the epilog on page 260, the story mentions briefly a "ring city" in the far future.
In this mention, the inhabitants of the ring city "opted for a permanent zero-gravity life" which clarifies that the ring city is not an orbital ring according to this answer's working definition.
Even the Orbital Rings around Trantor depicted in Season 2 of Apple's Foundation TV show look like they must make use of artificial gravity, so I do not think that these qualify either.
Obviously this may not be a complete list...