1
general::
Also, run a temperature. Suffer from a body temperature higher than normal, as in She was running a fever so I kept her home from school. These idioms use run in the sense of "cause to move," in this case upward. [Early 1900s]
American Heritage Idioms
2
general::
Phrase(s): run a fever [and] run a temperature
to have a body temperature higher than normal; to have a fever. • I ran a fever when I had the flu. • The baby is running a temperature and is grouchy.
McGrawhill's American Idioms And Phrasal Verbs