Before I knew what a fader was, I watched the sound guys at church put on their headphones at the beginning of the service and keep them on until the end, never or rarely listening to what the rest of us were hearing. They'd watch the meters, and the congregation had to live with something that might have sounded good to the sound crew, but didn't necessarily sound good in the sanctuary. That made absolutely no sense to me, so until a few years later, I wouldn't have been caught dead with a pair of headphones because no one ever taught me the value of using headphones correctly.
As a live sound mixer, you need a pair of headphones. There's no negotiating that point, but make sure that you use them the right way. Your primary job is to make sure that the system sounds good in the room, not to make sure that the board mix sounds good for the recording.
In the standard live sound set-up where you can hear the speakers directly, here are a handful of reasons to add a pair to your setup, if you don't own a pair already.
With a good pair of headphones, you can use the PFL (pre-fader listen) or Solo button on your console to listen to any one channel or group of channels.
Why might you choose to do that?
Have you ever needed to cue a CD, DVD, or taped accompaniment without everyone in the room hearing what you're doing? PFL means that you can listen to a channel in the headphones with the fader shut off. As a rookie sound mixer, I used to cue accompaniment tapes using only the meters on the cassette deck, and sometimes I cued things up to the wrong point. Not smart.
In the similar manner, a pair of headphones allows you to isolate a channel in order to find a hum, buzz, or noisy guitar amp. Make sure to mute the noisy guitar amplifiers during the sermon and/or use a noise gate.
With a pair of headphones, you can figure out who's singing off-key and keep them back a little in the mix, and you can tell if whether someone is really playing their instrument or singing. From 50-70' away, it's often hard to tell exactly what's going on. Sometimes, the keyboard player isn't really playing or a vocalist isn't singing.
When the guitar player wants more guitar in the monitors, you can quickly listen to his or her monitor mix to see whether more is needed or if you can achieve the same thing by reducing the level of something else that's covering up the guitar. It's a fact of life; guitar players always want more guitar. If you continually give everyone more of themselves, you'll soon have monitors that are too loud, which causes other problems.
A really good pair of headphones might set you back $150 or so, and there are plenty of good alternatives, even as low as $50. Whatever you do, if you don't have a pair of headphones and if your console has a headphone jack and PFL/Solo buttons, make sure that you get a pair. You'll wonder how you ever lived without them.
Take a look here for some great headphone options.