Naughty Dog isn't planning any DLC for The Last of Us 2.
Director Neil Druckmann confirmed this in a recent podcast with Kinda Funny. Druckmann, Ellie actor Ashley Johnson, and Joel actor Troy Baker joined host Greg Miller in a lengthy spoilercast on the game, and Druckmann plainly states that "there's no plans" for any DLC around the 1:52:00 mark. I haven't embedded the video because it's obviously loaded with spoilers, so if you haven't played or finished the game yet, you'll want to save it for a later date.
Many players have wondered if The Last of Us 2 would receive some sort of DLC add-on in the space of Left Behind, the DLC for the original game. Naughty Dog could've conceivably created another Ellie-centered DLC exploring any number of things, from her relationship with Joel to her growth over the years. But it seems the studio is more than happy with her story as-is, and quite frankly, so are we.
Nevertheless, Naughty Dog isn't necessarily finished with The Last of Us 2. As we reported last year, the multiplayer component that was originally planned for the game was cut because it "grew beyond an additional mode that could be included with our enormous single-player campaign." In other words, Naughty Dog is still planning an evolution of the Factions multiplayer mode established in The Last of Us online play, but it won't be part of The Last of Us 2 or a DLC.
Druckmann does mention Factions a bit later in the podcast - shortly after his remark about DLC, but again be mindful of spoilers - but only in passing. He doesn't really confirm anything, but his previous comment only rules out direct DLC, so it certainly sounds like the Factions mode is still A Thing.
These deleted scenes from The Last of Us 2 show a very different approach to Ellie's story - one which could have been used as a base for some sort of DLC.