Not every theme supports the “More” tag. Some themes are designed only to show an excerpt (such as Maranatha and Saved).
If your theme supports the “More” tag, try using Text mode (at the top-right of the editor) instead of Visual, or the
<!--more--> tag may not work. Read Excerpt or “More” Tag in the Blog guide for more information. This applies not only to blog posts but also sermons, events, etc.