One possible way is through a series of complex spells and potions. Harry, being the resourceful wizard he is, might spend months researching in the Hogwarts library. He could find a combination of spells that first weaken the connection of the Dark Mark to the Dark Lord and then use a special potion made from rare magical ingredients to completely erase it. This would involve a lot of trial and error, and he might even get some help from his friends Hermione and Ron who would assist him in gathering the ingredients and perfecting the spells.
In some fanfictions, Harry might use powerful ancient magic that he discovers in a hidden library or from an old wizard. This magic could be specifically designed to counter the dark magic of the Dark Mark.
One consequence could be that the Death Eaters no longer sense his connection to Voldemort. This gives Harry an advantage in his fight against them as they can't track him through the Mark.
Perhaps Harry uses some powerful ancient magic he discovers. Maybe he finds a spell in a long - forgotten book in the Room of Requirement that has the power to break dark enchantments like the Dark Mark.
It could change the way the wizarding world views Snape. Since the Dark Mark is a symbol of his connection to the Death Eaters, its removal might make people see him as a redeemed figure. For example, the students at Hogwarts who once feared him might start to respect him more. And the Order of the Phoenix members might trust him even more fully.
In this fanfic, Harry might have discovered a powerful magic spell or potion to achieve it. Maybe he found some ancient knowledge that gave him the means.
Well, if Harry has the Dark Mark in fanfiction, it would completely change his relationship with the wizarding world. He'd likely be seen as a threat and be shunned. His friends might turn against him out of fear or be forced to by the Ministry. It would also make his fight against Voldemort much more complex as he'd be associated with the Death Eaters.
The plot would change drastically. Firstly, Harry would lose the trust of many. Dumbledore might be more suspicious of him. Secondly, the Death Eaters could have different plans for him. They might try to use him as a weapon against the Order. Thirdly, Harry would have to constantly hide the mark, adding a new layer of secrecy to the story.
Perhaps in the fanfiction, Harry's connection to the Dark Mark leads him to discover a hidden side of Voldemort's past. This could involve uncovering secrets about Voldemort's family or the origin of the Dark Mark itself. He could then use this knowledge to try and find a way to break the connection or use it against the Dark forces in some way. For example, if he finds out that there's a certain spell that was used to create the Dark Mark and that same spell can be reversed, he might embark on a journey to find the ingredients or the incantation needed for that. And all the while, he has to deal with the prejudice and fear from those around him who might suspect him of being a Death Eater.
If Harry Potter gets the Dark Mark in fanfiction, it would completely change his relationship with the wizarding world. He'd be instantly seen as a Death Eater, and his friends would be in a state of shock and confusion. His standing at Hogwarts would be in jeopardy, and he'd likely be hunted by the Order of the Phoenix as well as the Ministry, at least until they figure out the truth.
Another plot could involve Harry being tricked into taking the Dark Mark. He might think he's doing something else, like a magical experiment or a ritual to save someone. But it turns out it was a setup by a dark wizard. Harry then has to face the consequences, not only the physical and magical changes brought by the Dark Mark but also the change in how others perceive him, especially his friends who might be shocked and scared.
Some of the die - hard Death Eaters could also be antagonists. They might not trust Harry even with the Dark Mark and try to undermine him or expose his true intentions if they suspect he's not truly on their side. Bellatrix Lestrange, for example, is so fanatical about Voldemort that she would be a likely candidate to oppose Harry even if he seems to be part of the 'dark' group.