Lance Wabisuke-Hamilton from One Piece: Parallel Works has a mysterious illness that causes him to fall asleep at random times and makes him physically weak. However, Word of God states that Lance's condition is not life-threatening.
Paul in the Pokemon fanfic Travels Of The Trifecta (yes, THATPaul) spends chapters 3 to 7 suffering from a severe cold/flu and later begins to cough up blood periodically. This is a sign that he has inherited the terminal illness that his mother died from.
In the Touhou Project fanfiction Reimu's Illness, because of her unstable (it is described as being up and down) health, Reimu falls ill with tuberculosis, human enterovirus 68, and viral meningitis and she is so sick that she's had to be transferred to a hospital in the city and be placed in something called an isolation box (which is akin to an incubator for premature infants).
In Gensokyo 20XX, she has been plagued with colds, depression, and various other ailments, even lack of sleep and appetite, requiring her to take so many medications to no avail. Ultimately, she was hospitalized and the doctors did all they really could.
In 20XXI, Eirin, due to stress, was starting to become this, developing some anemic condition as a result of stress, but she commits suicide before it becomes an issue
In 20XXII, Keine is revealed to have died of an illness consisting of "pneumonia and lumps in her lungs", Aya dies of heart-attack caused by illness and stress, and Koishi started to become this.
In 20XXIII and later on 20XXIV, after the nukes are dropped, Sakuya dies of radiation sickness.
In 20XXIV, Reimu's mother, Reiko, dies of a long term illness. Earlier, she was ill with radiation sickness before it dissipated.
Reimu (post age-regression) becomes this once again in Gensokyo 20XXV, after living in The Forbiddens for some time, and it was noted that she has been sick off and on.
From 20XXV, we have Yume Ni, in the aftermath of Chen's "The Reason You Suck" Speech. From what is said, she has a chronic illness and later on passed away.