The Naruto world has a rich lore with tailed beasts and different bloodlines. The Devil May Cry world has its own lore regarding demons and the underworld. In fanfictions that combine these two, the merging of these lores can result in really unique storylines. For example, a fanfiction could explore what would happen if a tailed beast was possessed by a powerful demon from the Devil May Cry world, and how the characters from both universes would deal with such a situation.
Maybe Naruto discovers some dark secret about Tsunade's past that she has been trying to hide. For example, something related to her lost loved ones. Naruto, in his usual straightforward way, confronts Tsunade about it without realizing how much it would hurt her, and thus makes her cry.
It could be that Sakura betrays Naruto's trust. For example, if Naruto entrusts Sakura with a very important secret, like something related to his family or his new power he's developing, and Sakura reveals it to their enemies or uses it against Naruto in some way, that would be a huge blow to Naruto and could lead to him crying.
In fanfictions, Minato might cry for Naruto due to the missed opportunities. He missed Naruto's childhood, all those years of Naruto being alone. When Minato sees the kind of person Naruto has become despite all the hardships, it can be overwhelming and lead to him crying. He regrets not being there for his son.
As a fan of web novels, I don't have any emotions and I won't cry because of a certain scene. But I can tell you some touching scenes in Naruto that might shock or move you.
Naruto worked hard to become Naruto and eventually overcame his psychological barrier to become a powerful Naruto.
2 Hinata had been silently supporting Naruto before he became Naruto and had put in a lot of effort and sweat for him.
3. In order to learn ninjutsu, Xiao Li continued to cultivate diligently and finally overcame his own difficulties to become an excellent ninja.
4. Naruto and Hinata lost their loved ones when Payne attacked Kinoha Village, but they still stood firmly on each other's side against Payne.
5. Erasu sacrificed his life to protect Sasuke and let Sasuke embark on the road of revenge.
6 Naruto and Hinata faced various challenges from the outside world and the inside world together on their wedding day, but they still supported each other and got through the difficulties together.
These scenes may shock or move you because they show the strength, courage, friendship and love of the characters in Naruto.
One possible reason could be the loss of a dear friend. Naruto forms deep bonds with his comrades, and if something were to happen to them, like Sasuke leaving again, it would break his heart and make him cry.
Naruto's ability to create shadow clones would be very useful. He could send clones to scout out areas full of demons, or use them as distractions in battle. His rasengan could be a powerful close - range attack against demons. Also, his chakra control could potentially be used to enhance Devil May Cry weapons, like making a sword cut through tougher demon hides more easily.
One possible plot could be that Naruto sacrifices himself to save the village from a powerful enemy. Hinata, who loves Naruto deeply, is devastated. She might cry as she remembers all their moments together, from when they were kids to their more recent interactions. In the fanfiction, she could then take on Naruto's will and strive to protect the village in his honor.
One interesting fanfiction could be where Naruto gains some of Dante's abilities from Devil May Cry. For example, he could inherit the power to use Devil Arms, which would make his combat style in the Naruto world very unique. He might use a Devil Arm sword along with his Rasengan, creating a whole new and powerful jutsu.
One reason could be extreme emotional pain. Maybe he witnesses the death of a very close friend or loved one in the fanfiction world. This overwhelming grief could cause such a physical manifestation as crying blood.