A reader of this blog kindly pointed out that my instructions for building Emacs 25.1 on Ubuntu 16.10 result in a core dump when the build process bootstraps emacs. I only tested the instructions on 16.04 so I hadn't run into this issue yet.
The core dump on 16.10 appears to be a known issue and there is a workaround. Instead of running configure with just the prefix argument, run it as follows:
The in-depth bug description is here in message #38. The short version is that gcc in Ubuntu 16.10 defaults to building code with -pie and this breaks Emacs. Turning off -pie results in a working executable.