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?
Leaving the LTE on will not drain your battery life. The data modem in your phone (a Qualcomm MDM9615M) is a chip that handles LTE and 3G signals. Therefore, your phone is not using a 3G chip and then draining your battery by having a second chip search for a LTE signal.
I know it's not specifically what you're asking, but you can disable LTE in settings.
Settings > Cellular > Enable LTE (switch to off)