Does leaving LTE on when LTE service is unavailable drain the battery faster? The iPhone 5-series phones all support LTE but if I only have a contract for 3G service will my iPhone battery drain faster because it's always trying to find an an LTE-capable signal? Can I stop it from even looking for LTE?
I know it's not specifically what you're asking, but you can disable LTE in settings.

Settings > Cellular > Enable LTE (switch to off)