First, thanks so much for your interest in Page-Links Plus and for your purchase.
As for the problem you describe, here’s a bit of background:
The PLP demo site (demo.pagelinksplus.com) always runs the latest WordPress nightly release and the most up-to-date versions of Page-Links Plus and the default WordPress theme. (At present, this is “Twenty Fifteen.”) The demo instance is also as “plugin light” as possible. In addition to Page-Links Plus, it runs only a handful of essential plugins, none of which interfere with one another or Page-Links Plus.
As such, if Page-Links Plus works in the demo environment, any issues purchasers face in their own instances can be attributed to plugin conflicts, outdated WordPress core files, theme problems (usually a missing <?php wp_link_pages(); ?> function), and/or other local issues.
In your case, since you’re running an up-to-date version of WordPress, I’d start with plugins and theme issues.
Second, if you verify that your theme includes the wp_link_pages functuion, try deactivating all your plugins (save PLP). If PLP works on its own, try reactivating the other plugins until you find the conflicting one.
If PLP still doesn’t work at that point, let me know.