I'm not aware of any such fanfic. Naruto is a male character in the series, and such a concept goes against the natural order of the Naruto universe. Fanfics usually follow the general theme and characteristics of the original work, and this seems quite out of place.
It could be the dominant character in the yaoi pairing. If there is a strong - willed character in the relationship, they might take it upon themselves to punish Kakashi for interfering or causing trouble.
Perhaps it's an organization that wants to use Kakashi for their own evil plans. They might think that Kakashi's skills and knowledge would be useful for them. For example, they could be trying to create some sort of super - weapon and they need Kakashi's expertise in ninjutsu.
Well, his friends might launch a rescue mission. They would use their various skills to track down where Kakashi is being held. Naruto could use his shadow clones to search different areas, while Sasuke might use his Sharingan to detect any hidden threats on the way to the rescue.
One way could be through words of wisdom. Kakashi has a lot of life experience and could share his insights with Sasuke, like how to deal with his inner turmoil and the path he should choose. For example, he might tell Sasuke that revenge isn't always the answer and that there are other ways to find peace.
In some fanfictions, it could be due to a special jutsu gone wrong. Maybe a jutsu that Kakashi was using to transfer some of his chakra to Naruto for a mission accidentally transferred his Sharingan power as well.
One possible plot could be that in the yaoi fanfiction world, there are certain unspoken codes or etiquettes. Kakashi, being unaware or simply ignoring them, does something that is considered inappropriate. As a result, his punishment could range from public shaming within the fictional community to being forced to make amends in a rather difficult or embarrassing way. For instance, he might have to publicly apologize to the characters he has wronged while wearing some sort of humiliating outfit.
If Naruto was raised by Kakashi in fanfiction, their relationship would be very different from the original. Kakashi could teach Naruto more about the Sharingan and its secrets since he has one. Naruto might also inherit some of Kakashi's reading habits and love for Icha - Icha Paradise, which could lead to some really funny moments.
One possible scenario could be that he is captured by an enemy ninja group who wants to extract information from him. They might use various cruel methods like chakra - draining techniques or physical harm. Another could be that he is tortured as a form of revenge by someone he has defeated in the past.
One interesting plotline could be Kakashi struggling to control the new power. Since the Rinnegan is extremely powerful, he might initially have a hard time adjusting. For instance, he accidentally activates its powers during a simple sparring session and causes a huge commotion. Another plotline could be how the other ninja react to Kakashi having the Rinnegan. Some might be jealous, some might be afraid, and some might see it as a new hope for the village.