Disabling the Google App
Disabling the Google App (com.google.android.googlequicksearchbox) may cause the Voice Input button on some standard keyboards (including the AOSP keyboard) to not appear or not work.

Some alternate keyboards, such as TouchPal, will allow downloading a separate speech-to-text package from Google.  Other keyboards, such as Swype, do not require any Google software to support voice input.

Outside of that issue, I have not found any consequences (outside of having more storage space) of disabling the Google App.