In this scenario, php returns a notice in which the $post object in wp-includes/post-template.php is null.
Notice: Trying to get property of non-object in wp-includes/post-template.php on line 209
Again, the notice throws only when no posts are created / published and you are attempting to request a custom post type (ie wp-json.php/posts?type=custom_post_type)
I have this problem aswell, and it seems to me that it stems from the prepare_excerpt() method in class-wp-json-posts.php. The global $post is not set when doing get_the_excerpt(), resulting in a function in post-template.php trying to use $post->post_content when $post is null.
I'm sorry I haven't had time to investigate further to why this is happening. I just merely commented out the call to prepare_excerpt() (we don't use the excerpt anyway) and went on with it.
Return null if post doesn't have an excerpt. Fixes #72.