The game doesn't check for rooted phones anymore, but it still does check for ro.debuggable, which is set for CyanogenMod and probably other ROMs. Also, it depends on Google Play Services (at the very least GCM and Play Store) and doesn't like being modified (clock error).
Vanilla Miitomo crashes on my CM13 Galaxy S3. When changing the isRoot, isEmulated etc. functions to “return 0”, it doesn't crash, but gets stuck with the time-related error (801-1650). I quickly searched through the files but couldn't find the codepath triggering this.
Edit: This might be...