In the movie, Joy is Riley's main emotion and cares very deeply about keeping her happy. When Joy is lost from headquarters, she desperately wants to get back so that she can keep Riley from feeling any negative emotions. While this may cause Joy to be bossy to the other emotions at times, she has good intentions and all of it stems from her care towards Riley.
In the film, Riley and Joy never really interact, as Joy is merely an emotion in Riley's mind. In earlier scripts of the film, however, it was intended for Joy (and the other emotions) to be able to come out of Riley's mind and interact with the girl regularly. Eventually, this was scrapped, and Joy became a permanent part of Riley's inner psyche.
Most fans prefer to keep their dynamic the same as it is in the movie, with Joy being Riley's inner voice and main emotion. Some fans take inspiration from the original scripts and have the two interact, but they are usually nothing more than good friends. A romantic interpretation between them is rare, but there may be a few fans who ship them.
Note: Please only add examples that are part of the RotBTD fandom.