Short answer: you should recompile
Long answer:
You don't have to recompile after changing the prefix from /usr to /usr/local with ./configure, but since emacs uses the prefix in pathnames embedded within the executable (such as /usr/lib/emacs/24.5/i686-pc-linux-gnu and /usr/share/emacs/24.5/etc; see src/epaths.h), some files that emacs reads at runtime will not be found, unless either
- you recompile emacs (recommended)
- before running emacs, you set and export environment variables such as
EMACSDATA, EMACSDOC, and EMACSPATH to override the incorrect builtin pathnames.
- you make symlinks in various places in
/usr to point to various places in /usr/local so that the incorrect builtin pathnames resolve to the new locations.
Running checkinstall to monitor the make install procedure and generate a .deb package comprised of all the installed files won't mitigate any of this; the emacs binary will still contain incorrect pathnames.