Is your nose is running an idiom?

"Your nose is running" is not an idiom. It's a literal description of what happens when you have a runny nose.

The idiom that describes having a runny nose is "to have a cold."

Here are some other idioms related to the nose:

* "To turn up your nose at something": To refuse to accept something because you think it's not good enough.

* "To keep your nose to the grindstone": To work very hard.

* "To poke your nose into something": To become involved in something that doesn't concern you.

* "To follow your nose": To go where your instincts tell you to go.