Well, it can vary a lot. Sometimes Barney is abused in terms of being mistreated in relationships or facing harsh circumstances that cause him harm. The specific ways depend on the creativity of the writer.
Barney's abuse in fanfiction could take forms like being bullied by other characters, facing extreme psychological stress, or enduring unfair treatment that affects his well-being. It all depends on how the fanfiction writer imagines and presents the story.
In some fanfictions, Barney might be abused physically or emotionally through various fictional scenarios crafted by the authors.