https://youtu.be/XAZewEpu56s?t=66
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Of course there isn't literal dialogue telling you to play all the classes in the game. However the systems in the game do highly encourage it by not only allowing you to do so on one character, but it rewards you for doing so as well.
You get bonus rewards on daily roulettes for in demand roles. Bonus xp when leveling a lower job than your highest one. Job specific mounts and titles that once achieved can be used on any job. Titles that can only be earned by leveling a group of jobs (God/Goddess of War/Magic). Gear is shareable with multiple jobs within the same role. Another strong encouragement is how FFXIV is a story focused MMO and you find out more about the story/lore by playing more jobs/roles. Speaking of the story, Venat, the previous Azem has multiple jobs. I'm sure I missed other ways of encouragement but I think it's pretty clear that it is there.
Even if these examples are somehow not strong encouragements, at the very least the game, outside of Savage raiding, doesn't prevent nor punish you for playing multiple jobs. If anything it enables you, while in Savage it pretty much deters you from doing so. At least until the content is just about obsolete.