Read more button not working

From the Codex:

Excerpts (teasers) can be shown on WordPress through two methods:

  • The first, keeping the the_content() template tag and inserting a
    quicktag called more at your desired “cut-off” point when editing
    the post.
  • The second, by replacing the the_content() template tag with the_excerpt().

If one of those conditions is met, and you are not getting the <!--more--> functionality then it is likely that your theme is doing something to bypass the the filters that make it work. Perhaps the theme is printing content via something like echo $post->post_content or the theme or a plugin has removed filters or added filters that break the <!--more--> tag. It is hard to say which without seeing the relevant code.