Does signing out turn Find My iPhone off?
You mean signing out of iCloud on the phone? Yes but you're turning the iCloud backup off too. You can always just turn Find My Phone off in the same screen.
If you mean signing out of icloud.com no, you would need to go to the device at icloud.com under Find My Phone. Devices, and x out to tell it to forget the phone. That should flip the Find My Phone switch on the phone to off, you need to be connected to wifi on the phone though for it to work.