Healing hasn't gotten harder, per say. Just parties don't collectively outgear content by 20+ ilvls anymore. Everyone is back to leveling (and gearing) up, so tanks can't always go balls to the wall in dungeons anymore, pulling everything between bosses and not using any mitigation for it. Parties have to adjust to the fact we aren't putgearing content anymore. Heavensward was similar in regards to its leveling dungeons, so this isn't really a new thing. With time and higher ilvl gear, I'm sure Stormblood content will devolve into what Heavensward content eventually became in terms of outgoing damage and healing requirements.

Addition: and from what I've heard, tank gameplay has changed so there's probably a lot of tanks still adjusting to the new playstyles of their jobs. I haven't touched any of my tanks yet, so I can't speak on specifics here, only repeat what I've heard.