"running a fever" origin

I'm not sure of the actual entymology of the phrase, but 'to run' means in this case 'to continue to have', so while the fever is being maintained it's 'running'.

I always considered this to be something of a misplaced ownership of the word running.

It is not the person who is running a fever but rather the person has a fever and it is the fever that is running, running it's course; eventually the fever will complete it's run and the person's temperature drops to normal again (or they are dead).

Ngran shows that its usage started to become popular at the beginning of 1900.

I have found the following reference :to run a fever

Etymology:

'Run' has many meanings. In this phrase, 'run' means 'to continue to have', so 'run a fever' refers to a person maintaining a 'fever' (a high body temperature).