I've heard a lot of raving about this series now and I really want to pick it up. I'm struggling with some pretty heavy depression however, so I don't know if I can handle more bleak and depressing than real life already deals me.

I've seen some comparisons being made to Dragon Age: Origins and GoT, but I have no problem with those. The former because a) how dark the story gets is mostly up to what choices you make, and b) despite all the tragedy going on, there are moments of fun and camraderie that makes up for it.

The latter because... well, I don't know why, actually. Probably because the tragedy in GoT feels purposeful and plot-relevant somehow, as opposed to it just being "look, fiction sucks as much as reality, so people die and suffer all over the place because of it". (I had more trouble stomaching DA2 because of that; that was basically "Everything Goes Wrong in Hawke's Life No Matter What S/he Does: The Story".) And even then the characters that live on still have fighting spirit in them, like everything that happens just make them stronger people. Or maybe it's because I read all the GoT books a while back before the depression really kicked in, I don't know.

TL;DR: Should someone battling depression play this game/series, or set it aside for later?

The Witcher isn't REALLY dark. The books are a bit more bleak than the games. In the first game, I hadn't read the books and didn't realized just what a looming threat Nilfgaard was or how depraved the Scoia' tael were. The game has lore, and if you read that it's still not oppressive. Geralt really feels like a one man army, but he's usually successful at what he does in seeking out his cause. In the first game, that's Salamandra.

I played TW1 up to the end of chapter 2 (quit for boredom) and the whole TW2. They're definitely much darker than Dragon Age (I don't watch/read GoT), and have a rather bleak vision of humanity. I'm not struggling with depression - not anymore at least, and by the way, best of luck with that, I mean it - but I must say it did bother me a bit.
It's a world in which most of the people you meet (as in, named NPCs) will probably surprise you negatively, and are hiding more than one skeleton in the locker. The common folks are pure trash (this is also evidenced by their look: everyone's deformed and ugly except for your friends and a few other NPCs) their lords and masters are even nastier, and you're no saint either, you're a mercenary whose main concern is money and <insert main quest here>. Geralt himself is kinda neutral, he's not heartless and he doesn't like cruelty and injustices, but he knows it's not up to him to save the world. Without getting into details, let's just say that even if you played your Geralt as a charitable good guy, things might not work out as you expect. Moral choices in the Witcher tend to range between 'oh that's too bad' and 'shit happens'.

I can't say whether this is something you can like or not, but Witcher 1 has a demo (which, if I recall correctly it's the whole prologue + chapter 1) so I recommend playing it to get an idea of what it feels like. If you decide to play them anyway you're probably better off starting with TW3 though, or at least skip TW1 entirely.

I've only played 1/3 to 1/2 of the first game (and nothing of any of the sequels/spin-offs), but from what I've seen so far, it's certainly no darker than A Song of Ice and Fire -- though I've only read the books, so I can't say how those compare to the Game of Thrones TV series. The entire Witcher series, though, promotes itself as being not about right and wrong, but about choices and their consequences.

If you're worried about the tone of the game negatively affecting your mental well-being, though, I'd say maybe just stick it on your wishlist for now; maybe revisit it in the future when you're in a better state of mind. Also, the game gets discounted heavily pretty often, so if you got it for $2 US (or the equivalent), you wouldn't be out much if the game wound up being too bleak for you.