Who is supposed to use limit break and when?

[Zetsumei_Tsunarashi]

The only time it's a good idea to use bard (healing) limit break, is when you have level 3 and most of the party is dead. Any other time it's a waste.

In general the melee DPS will be the one to use limit break and it will be near the end of a fight in order to finish off the boss. However, depending on the fight, limit break can be used as part of a boss strategy. For example on ifrit HM, BLM will often limit break during the nails phase in order to kill all the nails quickly, or during garuda HM to kill the sisters quickly.

In most cases, limit break will be used to do damage. Tank and healer limit breaks are normally only used as a last resort.