run a fever
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]
The American Heritage® Idioms Dictionary Copyright © 2002, 2001, 1995 by Houghton Mifflin Harcourt Publishing Company. Published by Houghton Mifflin Harcourt Publishing Company.