In the grand scheme, whether you monitor or not depends almost entirely on two things...first is whether your rig is new to you or not. If it's new to you and you're uncertain about such things as settings, levels, etc then monitoring is obviously recommended so you can correct anything that's wrong before it's too late. Once you get confident of your setup, you probably won't need to lug your headphones around with you because all you need to monitor during the show is levels to have a high level of confidence that you're getting a good quality recording. From my experience, most tapers know their gear well enough that they don't need to monitor through headphones.
The second factor, as Dan Lynch suggested previously, is if you have alot of variables in your recording scenario that you'd want to monitor the output to ensure that nothing is outta whack. Once the music starts, if there's lots of tweeking going on because, for example, you're running 4 channels or you're doing mid-side and mixing on the fly, they you probably wouldn't want to trust fate and find out afterwards that you could have fixed something had you been monitoring.