If you're a game developer, journalist, or just enthusiastic about the topic, you should play DF. You don't have to play it a lot, and I'm not saying you'll love it. But at this point, DF has a relationship to modern games much like a literary classic does to other books: it will help you understand where a lot of stuff is coming from.
Developers don't talk as much about their influences as I think they should. I guess there's a worry you'll be accused of being derivative, but ideas don't come from a vacuum after all. I would love to see a family tree of games with each influence nicely mapped so you could follow how ideas emerge, combine and refine over time. And in DF's case, you would find it a tight knot of influences.
It's well-established that Minecraft was heavily inspired by DF. In an early conception, it was meant to be a strategy/management game, basically Infiniminer crossed with DF. The game soon became first-person, but many influences remained. Minecraft in turn has had a broad influence, both in terms of direct clones (none of which did very well), and in influencing games that took a particular part of the Minecraft experience and focused on it, such as Don't Starve.
Most directly, DF has inspired a whole lot of similar games that Full Glass Empty Clip has dubbed "Dorflikes". These range from very similar games like Gnomoria or the Ill-fated Towns, via games like Banished, to Rimworld and even Prison Architect. That is not to say that DF invented the genre of indirect management games, as you could call Dorflikes - games like The Settlers were there long before and presumably inspired DF.
Now I may have mentioned previously that I don't think Dwarf Fortress is actually a very good game. It seems incredibly difficult at first, but nearly all of this difficulty is in figuring out the weird and inconsistent interface. I have definitely written before about how I want my games to "push back", to remain difficult enough so that my victories feel real and hard-won. DF, like many of its successors, fails in that drastically. I am by no means an expert or obsessive player, but I generally find that as long as I give my dwarves one axe and one pick and drop them off somewhere that isn't a complete hellscape, they'll do just fine. Never mind that - I've had successful fortresses built beneath glaciers haunted by undead yetis.
A big reason for this lack of pushback is the sheer depth and variability of the game. How do you balance a game that is this complex? Another is the indirect control you exert over your dwarves. One way in which the game is absurdly easy: the dwarves only eat a tiny amount of food. A single semi-competent part-time farmer can sustain a dozen of them with not much trouble. But if the food requirements were tighter, the fact that your dwarves will sometimes just get themselves stuck in some mess of competing requirements and resource shortages would mean that pretty much all fortresses would quickly succumb to starvation.
But enough complaining. Dwarf Fortress is still a fascinating game because there's just so much of it: resource management, 3D construction, automatic generation of worlds and stories, a frighteningly detailed combat system, creatures, geology, and a whole adventure mode I haven't even touched on.
Games derived from DF inevitably take one or a few of these things and concentrate on them, not unreasonably. This means that much like with a seminal piece of literature, it's still very much worth going back to find different aspects to expand upon. In a sense, this is sad: DF, rather than becoming as wildly successful as its progeny, gets strip-mined for ideas. But that's pretty much how culture works: the most popular version of an idea is almost never the first, or the most intricate.
And there's still so much to find: No other game so bravely tries to create a different new world with a different history for you each time you play. The need for polish and "assets" means that most games' stories are still utterly linear, their worlds not living things but a series of flimsy stages held together by plot.
And as much as 3D graphics are dominant these days, in terms of semantics, games tend towards the 2D. This is one of the things that makes Minecraft so engrossing: worlds that are not facades but physical shapes with volume. You can have a dozen rooms above one another, and jump from a window to a tree, climb the tree and jump back into a window above - not because a level designer considered and designed this path, but because it happens to be there.
In big games, the capacity for level size and complexity is forever leeched by the need for fancier graphics, the demands of a linear plot, and the desire for scripted setpieces. So play Dwarf Fortress, play Minecraft, and think about mining them some more for their ideas of physical, interactive worlds.
For now, of course, I'll just be working on Airships: Conquer the Skies...