The concept changes the relationships around Harry too. His friends may be more cautious or in some cases, more protective. In a fanfic I read, Hermione spends a lot of time researching how to help Harry control his Obscurial side. Also, the wizarding society's view of Harry could be different. They might fear him more because of his Obscurial nature, thinking he could lose control at any moment and cause destruction. This adds a whole new layer of tension and drama to the fanfic universe.
In many Obscurial Harry fanfics, it alters the power dynamics at Hogwarts. Harry may not be just the Boy - Who - Lived but also a potential threat. Teachers like Dumbledore have to approach him differently. They need to balance between helping him develop his magic and keeping his Obscurial side in check. This leads to some interesting plot developments where Harry might be isolated or grouped with other 'dangerous' students. It really shakes up the typical Hogwarts experience we know from the original books.
Well, in fanfics, Obscurial Harry often means a more unpredictable Harry. His power can be a wild card. For example, in some fanfics, it makes his battles against Voldemort more intense as he has this hidden, often uncontrollable power.