When you add an image for a forum post using the HTML editor, this image is not visible on the email notification of this post because the url of the image is restricted to logged users.
This old issue (MDL-42007) has an implementation of a solution, where the it adds the image to the mail before sending it.
This issue is said to be duplicated by MDL-41956 which is now fixed in Moodle 2.6.

The problem is that I think MDL-41956 solved other problem (attachment files but not embedded images), but forgot to solve this particular one. I've tried with Moodle 2.7, but the forum post images were not embedded to the mail. So either there is a setting for that and I didn't find it or this was actually not added in MDL-41956. But after having a quick look at the code, I think it's still not solved.