ASSALAM U ALAIKUM
Is it permissible to continue to keep fasting with high fever which causes difficulty in not taking timely medicines?
When is it permissible to break ones fard fast in sickness and how can one derive the tipping point of when to do break it?
In the Name of Allah, the Most Gracious, the Most Merciful.
As-salāmu ‘alaykum wa-rahmatullāhi wa-barakātuh.
In principle, if a person has such a sickness that fasting will either increase the sickness, delay the cure of the sickness, or cause death, then such an individual is excused from fasting. However, this information must be attained by consulting a practicing Muslim doctor.
If one is fasting and during the course of fasting he/she experiences such a sickness that continuing the fast will either cause death or cause the sickness to increase, then one may break the fast. 
In both cases, one must do Qadha (make up) of the missed and broken fasts.
And Allah Ta’āla Knows Best
(Mufti) Abdul Azeem bin Abdur Rahman
 Beheshti Zewar (Urdu), page 233, Altaf and Sons