Q: Does the phone need to be turned on in order to be located by the "find my iphone" application?

Yes, it needs to be turned on, and connected to a WiFi or cellular network.

And it needs to have

1.  The  "Find my Phone" switch turned to "On" in iCloud AND

2.  The location services turned on Settings>privacy> Location Services=ON

Before it is misplaced.

While this may seem intuitive it seems a lot of folks miss these steps.

Cheers