One example could be a story where Beast Boy gets injured during a battle with a powerful new villain in the city. He might struggle with his powers not working properly due to the injury, which adds a lot of drama to the fanfiction.
You can start by looking on popular fanfiction websites such as Archive of Our Own (AO3). There are likely to be many different stories there that involve Beast Boy getting hurt in various ways. Just use the search function and type in relevant keywords like 'Beast Boy hurt'.
Perhaps there is a magic user who wants to capture Raven for her powers. Beast Boy tries to stop the magic user and ends up getting hurt. This could be a situation where the magic user casts a spell that Beast Boy tries to absorb or deflect to protect Raven. His injury then becomes a significant part of the story as Raven has to figure out how to save him and deal with her feelings for him at the same time.
In many Beast Boy and Raven fanfictions, Beast Boy's act of protecting Raven is quite common. When he gets hurt in the process, it often adds a layer of drama. Maybe he takes a blow meant for Raven during a battle with a powerful villain. This could lead to Raven realizing her deeper feelings for him as she tends to his wounds.
To write a Beast Boy hurt comfort fanfiction, start with a strong setup for the hurt. For instance, Beast Boy could be betrayed by someone he thought was a friend. This would be a great emotional wound. Next, think about how you want the comfort to be delivered. Maybe Beast Boy runs away and hides in the forest. Raven, being able to sense his distress, follows him. She doesn't say much at first but just sits beside him. As time passes, she shares her own experiences of feeling alone and how she overcame it. This slowly helps Beast Boy open up and start to heal. Include lots of description of their emotions and the setting to make it more engaging.
In some fanfictions, Beast Boy might get poisoned while on a mission. Starfire would then use her alien knowledge to find a cure. As she's taking care of him, she tells him stories of her home planet to keep his spirits up. The story could focus on how Beast Boy is physically in pain but also comforted by Starfire's presence and her tales, which give him hope and a distraction from his suffering.
The story could end well if the Titans manage to defeat the cause of his injury. For example, if it was a villain, they could capture or even rehabilitate the villain. And Beast Boy, through his experience, could grow closer to the other Titans and they could have a big celebration once he's fully healed.
She would likely feel guilt. Since Beast Boy got hurt protecting her, she would blame herself to some extent for putting him in that dangerous situation.
One idea could be that they are hurt during a battle with a powerful new villain. Maybe they underestimate the enemy's strength and end up with physical injuries. This could lead to them having to rely on each other more for support and also force them to confront their own weaknesses.
One possible plot development could be that the other Titans rush to find a way to heal him. Maybe they have to seek out a rare herb or a powerful magic user. Another could be that Beast Boy's injury causes him to have strange side - effects, like losing some of his shape - shifting abilities temporarily. And it might also lead to some self - discovery for Beast Boy as he copes with his pain and weakness.