When to use Hellfire High Council Legendary Ring Effects

The DPS legendary ring effect should ideally be used during each boss's "final hour" phase, when they are below 30% health and gain additional abilities. However, in a typical raid, Jubei'thos will reach his final hour shortly after Gurtogg has died (because the two bosses are tanked together and Jubei'thos will take cleave damage). It is more important to use the legendary effect on Gurtogg's final hour phase, as players have to deal with its lingering effects for the rest of the fight.
The healer legendary ring effect should be used during Wailing Horror/ Darkness phases, and at the end of the fight when Dia Darkwhisper reaches 30% health.
There is no particularly strong time to use the tank legendary ring effect; during Nightmare Visage, especially at the end of the fight, or if the Gurtogg tank fails to reset Acidic Wounds, are both reasonable uses.
Looking For Raid Difficulty
On Looking for Raid difficulty, Dia's Void Haze mechanic does not exist. As a result, many raids will tank all three bosses together, and since Dia has the lowest total health pool, she is likely to die second with a group attempting to follow the correct strategy, or may even die first in an inexperienced group.
Healers should be prepared for this; the fight becomes very taxing on healing and mana when Dia is dead and the entire raid will be affected by Mark of the Necromancer. The Mark should still be dispelled after this time, as it will reset the debuff to its low-damage variant.