How do I remove contact photos from the messages app in iOS 10?
There is no way to change the layout of the Messages app to not display photos at the top of the screen.

If you go to your list of conversations and then tap on one, as the screen transitions you can see the difference in the height of the navigation bar at the top that contains the photo. (The conversation list uses a standard-height navigation bar that should be identical, or within a few pixels, of the height in iOS 9.) I took screenshots and measured - it's 42 pixels. Using normal-sized text (as set in the Settings app) that's approximately one line of text.