If you're comfortable (and what I'm about to suggest probably isn't the best security advice), you can do what we've done and open a port on your router (assuming you have a static IP) and redirect it to the OpenLP machine (which will also need to be static on the network).
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We've then done an HTTP redirect so that words.ourchurch.com redirects to our church's IP and it shows people the live view. Not that we've actually used it with a whole congregation but our tests have gone well - it only lags behind by about a second.
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The computer we have OpenLP on is used solely for that purpose and houses no sensitive info, probably not best if it's also used for other things.