DO YOU NEED TO DPS AS A HEALER : Absolutely not.

And her are a few more examples.

Do you need to use cool down as a tank? No.
Do you need to use heavy thrust as a dragoon? No.
Do you need to get greased lightning stacks as monk? Nope.
Do you need to use ninjutsu as a ninja? Not at all.
Do you need to use all the skills available to you on any class? Nope.

Will using ALL the skills available to you be beneficial to you and your party members, increasing your chances at clearing new and difficult content? The answer is a resounding YES! Absolutely.

This game is quite far removed from healers healing, tanks holding the boss down while the dps beat it.

Healers can DPS to assist in overall damage, and push phases predictably.
Tanks can cycle cooldowns logically and use several self healing abilities.
DPS can also use self healing abilities and assist the healers by putting up defensive buffs on the party and tanks.