I already own the base game of The Witcher 3 from GOG, but just happened to get The Witcher 3 GOTY in a Mystery Bundle from GMG.

When I add the GOTY key to GOG, will TW3 update with the DLC or will I have to redownload the entire game?

Also, is there a way to give my existing base copy of TW3 away or will it just be wasted? (I'm reluctant to add the GOTY key for this reason)

You gonna have two separate games in your GOG library. Since your base game doesn't have expansions, to access them you'll have to download and install entire GOTY. Sucks but it'll take a couple of hours with modern internet speeds ;)
As far as giving away your bases game: it's DRM-free, so do with it as you will but I don't think GOG will transfer it out to someone else's account at will, unless you buy it expressly as a gift. Might be worth shooting an email to support but I wouldn't hold my breath for it. Or just gift the GOTY key to whoever and buy the expansions for yourself, but you'll end up with two W3s in your account again lol.

It's possible I suppose, as it does work the other way around.
I owned the base game and season pass, and now I have two entires in my library. One for base game and DLC's separate, and one for GOTY edition of the game.
Don't know why they did it this way, but it might mean you are in luck.

Why you would want it though if you have GOTY is another story

Thanks for your advice. I'll contact support and see what happens.