I don't know the story, so maybe the fox died on the run. However, "On the run" does not mean hopelessness or death: that depends on what happened next to the fox. You are quite correct about the meaning

"The fox on the run" is hunted by hounds. This is a very unfortunate ending!

You're right that it doesn't imply any specific outcomes. It sounds to me like "fox on the run" is actually referring to the "she", not to the "boy". it's describing her as someone who perhaps doesn't like to stay in one place or remain with one person, but prefers to keep moving. Edit: That makes a lot more sense, Joey! Thank you. Context is everything.

She walked through the corn leading down to the river. Her hair shone like gold in the hot morning sun. She took all the love that a poor boy could give her And left me to die like the fox on the run. Like the fox (like the fox, like the fox) on the run. Everybody knows the reason for the fall When woman tempted man down in paradise's hall. This woman tempted me all right, then took me for a ride But like the lonely fox I need a place to hide. Like the fox (like the fox, like the fox) on the run. Come take a glass of wine and fortify your soul. We'll talk about the world and friends we used to know. I'll illustrate, a girl put me on the floor. The game is nearly up, the hounds are at my door. Like the fox (like the fox, like the fox) on the run.