We say, "I'm running a fever" or "I have a fever" to say we have a fever. To say that the fever is gone, we say, "my fever broke" (often followed by a time, as in, "My fever broke two hours ago, so I'm feeling much better"). You can also just say, "I don't have a fever anymore."

"I am running out of fever" does not make sense. It should be; "I am running a fever"= I have a very high temperature.

Never heard anyone say that. Much more common is... "I am running out of money/patience/time etc. I am running out of fever sounds like you are complaining! I am running out of fever... I want more!