On bootup sometimes fsck cannot fix everything it drops the user into some kind of maint mode and reports to the user that fsck needs to be run without flags. fsck should always give a very stromg warning that it is dangerous to run on a mounted file system if that attempt is made. fsck called me a probable "bonehead" after it had already made some changes. I laughed, but wished it had stated it earlier and informed this "bonehead" how to fsck the root filesystem without mounting it. Fortunately no damage was done and I have a sense of humor about the incident.
fsck'ing a read-only mounted file system (like the root filesystem in the emergency mode you're referring to) can't hurt.