Preheat a 6-quart air fryer to 350˚ F. Sprinkle the ribs with the chile-lime seasoning. Arrange in the air fryer basket in an even layer (it’s OK if they fit snugly). Cook, turning halfway through, until the ribs are browned and crisp, 25 to 30 minutes.
Meanwhile, heat the olive oil in a medium pot over medium-high heat. Add the onion, celery, carrot and garlic; cook until the vegetables are tender, 6 to 8 minutes. Season with 1/2 teaspoon each salt and pepper. Add the collard greens, vinegar, sugar and 3/4 cup water and bring to a boil. Reduce the heat to a simmer, loosely cover and cook until the collards are tender, 20 to 25 minutes. Season with salt and pepper, if needed.
Transfer the ribs to a large bowl; add the barbecue sauce and toss to coat. Return the ribs to the air fryer and cook until the sauce is slightly sticky, 5 minutes. Serve with the collard greens, rolls and more barbecue sauce.