Is there a word or phrase for running away from your wife or family because of their illness?
Solution 1:
"Abandonment" might fit. It's the U.S. legal term at least.
From the New Jersey state legal code:
2C:24-8. Abandonment, neglect of elderly person, disabled adult; third degree crime
- a. A person having a legal duty to care for or who has assumed continuing responsibility for the care of a person 60 years of age or older or a disabled adult, who abandons the elderly person or disabled adult or unreasonably neglects to do or fails to permit to be done any act necessary for the physical or mental health of the elderly person or disabled adult, is guilty of a crime of the third degree. For purposes of this section "abandon" means the willful desertion or forsaking of an elderly person or disabled adult.
Solution 2:
Consider ditch
End a relationship with (someone) peremptorily
[Oxford Dictionaries Online]
Usage:
Mary's husband ditched her because he no longer could take care of her dementia related illness and related expenses.