Does Dark Souls III follow Dark Souls and Dark Souls II in the storyline?

Basically; no.

While the story might share similarities through the references of "the old gods" and the "chosen one", with perhaps a few old faces thrown in, the story of DkS1, DkS2, and now DkS3 are not really related.

If you want to follow the Lore, and the history of the series, then by that standard, you may need to play the older games to follow that through, but there are a lot of YouTubers that already have covered a lot of the DkS universe.

EDIT: to slightly contradict my original answer, the community is (unofficially) calling DkS3 the "1.5" of the series. There are similar areas like the Hub and Anor Londo, and a few old faces thrown in, but the primary storyline is very different to the other two games - focussing on the themes of fire, rather than darkness.

Again: no.

My previous answer was actually without any involved experience of the game (YouTube mostly), so, as stated in the comments of the answer, it was very much a "shortcut answer". However, having now actually played the game in depth, and explored the lore a fair bit more, I can elaborate further.

So, no, Dark Souls 3 is not a "direct" sequel of the previous games. There are familiar faces, areas, and items that you might recognize if you play any of the previous games, including, once again, the ever-present Moonlight Greatsword.

To divert from my original answer though, Dark Souls 3 does set the theme of "the end". While yes, there are faces form both Dark Souls 1 and 2, even though only areas from 1 are present ("Time is convoluted") - the theme starts off as "this has been done again and again... linking the flame over and over, and everyone is just getting tired of it."

But, once again, the fire must be linked, and as stated previously, themes of fire, rather than darkness are more present, but the mindset is generally "Bored of this cycle. I'm going to get takeout. Be back never", and your job is to force them to do it, or find an alternative.

Comparing this to earlier games, the "cycle" appears to be dwindling over time. The ending of 1, next to 3, are very different, and honestly, 2 is just appearing to be filler, the more I look at it (no one is really trying to do anything about the cycle, they're just looking at ways to fix the symptoms, not the cause).

So to answer your question: no, you don't have to play any of the prequels, if you're just looking to play a good game. However, if you want to figure out what's going on, the rabbit hole will likely lead you to the previous games regardless.