Bottom line: if you are not DPSing as a healer at all, you are not playing your job optimally and you are not doing 100%, everything you can for your group.

Of course it's not "required" (in most cases), your primary function is to keep people alive. :P And if you're fine doing the bare minimum you have to do without any drive to excel or be an exception player (and thus, an exceptional healer), then sure, don't worry about DPS, as long as you're keeping everyone alive. Twiddle your thumbs when healing is not needed. Overheal for giggles/keep everyone super topped up all the time (as long as you're not making the tank's life miserable). Whatever. In the end, it's how you want to play, so long as you aren't hindering the progress/play of others.

You will never be an exceptional healer. You'll be ok, but you won't be able to be considered anything beyond that.

That is my take on it. And if I do get into a party with a healer that refuses to DPS, that's fine. If they were clearly new to healing I used to give them the little spiel (I WAS nice about it, mind you) about how DPSing as a healer allows you to get the most out of your role/job and provide the most utility for your party, but no matter how nice I was about it, people continued to get offended. So now I don't bother. Instead, I figure they've got things handled in the heals department, and I can push MY DPS numbers even higher. (Yes, I'll help where it's needed. I'm really not an asshole.)

TL;DR: If you're fine with, for all intents and purposes, being a mediocre healer, don't DPS. It's okay. It's not "required". If you want to grow beyond that, well, you're going to have to DPS. :P