You are never required to DPS as a healer.
You will however feel bored at times when the party actually knows how to avoid damage. The best point of reference is Castrum Meridianum, as there will be two tanks and two healers, and you may end up with 4 DPS that may downright suck. If you can get through a run without anyone dying, then you can probably find the windows to DPS in. Certain dungeons I'd probably not recommend ever DPS'ing (Aurum Vale for one) because the healing requirements are excessive.
I think the opposite is true. DPS'ing feels like a waste of time on healer because an entire switch to Cleric Stance lets you fire off any two DPS attacks or one Holy in that 5 second window before you can switch back. Holy does maybe 600 Damage (900 Crit), while a BLM casting pretty much anything does more damage and they have the same DPS cast speed but Astral Fire winds up making them do 5 times as much damage. As a point of reference, WHM can take Blizzard II from THM, but even with Cleric Stance, it does much weaker damage. In all the dungeons that you can't use Holy, DPS'ing as a healer lets you switch to Cleric stance, apply the DoT's, and then switch back.
There's rarely a situation where the Healer's DPS'ing makes a difference, since you're not contributing 25% of the DPS, you're contributing maybe 3% at most. The only time that it makes any real sense to DPS is when no damage is being taken, and won't be taken for a long enough window. Syrcus Tower has one phase like this during meteorfall. Ultima Weapon has a phase like this (though usually the Limit Break is fired here.)
That has never happened. Likewise, most Bards never use their Song skills, and nobody uses their self-healing skills. It's rather silly to think that the Healer should be contributing to DPS while the rest of the party does nothing to help heal themselves or... avoid taking damage entirely.
If the Healer "has to" DPS, then that is a tactless admission that you think the DPS are not pulling their weight. If a DPS casts Cure/Physick on another player, that's the DPS tactlessly admitting they think the Healer is awful. Similar statements can be made about players casting Stone Skin or Protect that aren't the healer. (Keeping in mind that it's not available to CNJ/WHM before level 34)