Maybe Mrs Black made some offensive remarks about Harry's friends or his family. For example, she might have insulted Hermione for being a Muggle - born witch, which would definitely make Harry angry enough to shout at her.
The Weasley twins could be part of it too. Suppose Harry yells in excitement or anger about a new prank idea that the twins have come up with that might be a bit too dangerous. They could either encourage his yelling or try to tone it down, depending on their mood. And Malfoy might overhear Harry's yell and use it as an opportunity to taunt him, adding more conflict to the fanfiction.
It could also cause a stir among the portraits in Grimmauld Place. The other portraits might side with Mrs Black and start to give Harry a hard time whenever he is in the house. Or, some portraits might be secretly impressed by Harry's courage and start to question their own long - held beliefs.
Maybe Albus has done something dangerous or disobeyed Harry's instructions. For example, if Albus got involved with some dark magic related stuff without Harry's knowledge, that would surely make Harry yell at him.
One possible reason could be that Mrs Black's old - fashioned pure - blood views conflict with Hermione's more inclusive and progressive ideas. Hermione, being a Muggle - born witch, might not tolerate Mrs Black's snobbish and discriminatory remarks about non - pure - blood wizards and witches.
Maybe she's yelling at Ron. Their relationship in fanfictions can be quite complex. If Ron says something insensitive or does something stupid, like forgetting an important task that could put their mission in jeopardy, Hermione would surely yell at him.
Perhaps in a 'harry potter yells fanfiction', Harry yells when he discovers that someone has been spreading false rumors about his friends. He could be in the Great Hall, stand up, and yell at the perpetrator, creating a big scene full of drama and conflict within Hogwarts.
Mrs Black could be so offended that she starts to plot against Hermione. For example, she might try to get Hermione into trouble with the Hogwarts authorities by spreading false rumors about her.
Maybe James Sirius Potter did something really reckless. For example, he might have put himself in danger while exploring some forbidden areas in Hogwarts. Harry, having gone through a lot of dangerous situations himself, would be worried and yell out of fear and frustration.