There's no facility within Lightroom Modile to do this. From Lightroom Desktop, you can use Jeffrey Friedl's excellent LR Publishing Plugin to allow you to post to a business page, but if you're trying to post from your phone, you'll have to do the two-step dance of saving to your camera roll, and uploading from there within the Facebook app.
Here's a link to the plugin for desktop:
Mike, thanks very much for your helpful answer. Hopefully, Adobe will implement this in this future. But, in the meantime, I will check those options out.
All the best,
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The problem is that facebook hasn't implemented the service in their "pages" app. They have in their normal facebook app but not in the app that you use to manage pages. They could easily implement this and Lightroom would automatically gain the capability to share to a page as Lightroom mobile uses the standard way sending files between apps works in iOS.
still not working?
Share it to Instagram and connect your Facebook Business page to Instagram.
So when posting from Instagram you should be able to pass it through to Facebook as well.