Generally LB priority goes like this:

Bosses:
Melee LB --> Caster LB --> Healer LB --> Tank LB
Melee use their Limit Break at an opportune time to either take a chunk out of the boss's HP or to finish it off. EXCEPTION: Ifrit EX has the melee LB the large nail often.
Casters use their Limit Break if no melee are available or if a damage check needs to be passed fighting multiple enemies.
Healer LBs are used at level 3 to prevent a full wipe and recover from a bad situation. Generally these aren't used before that.
Tank LBs when there's really no other thing to use the LB for. EXCEPTION: Ultima HM, when tanks are the ONLY ones LBing.

Dungeon Mobs/FATE:
Caster LB --> Melee LB --> Don't LB lol
Caster LBs destroy mobs.
Melee LB only if there are no casters to LB, or if there's a big monster.
Don't LB otherwise, waste of time/LB meter for incoming bosses.