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Who wins

I think there are way too many fighters with way too many options for killing Kurse (like, say, the killing curse )

For reals, Warcraft and Potter verses find a way between the Fell, spells, etc.

Khadgar turns him into a sheep.

I guess I'll ask the obvious answer, can they use magic against beings with (for the sake of the argument) infinite durability.

Kurse doesn't have infinite durability though. As pointed out in another thread, Loki stabbed right through him, using an Elven sword. Another Asgardian also managed to cut into his shoulder, during the prison break scene. The fact that he ignored it simply means he has great pain threshold/damage soak, but it also shows he can be physically harmed.

Also, he did great against conventional attacks, but what feats does he have against exotic magical powers? He certainly wasn't immune to falling for Loki's illusions. To assume he has full magical immunity, based on taking physical attacks, is a massive no-limits fallacy.

I was asking that same question about the killing curse...

Well, it wasn't the most specific wording to a question. And there are also other spells, like bewitching sleep, from the Potterverse, that can work as a KO. Or Gul'dan's life force drain, which doesn't attack you physically, but literally sucks the life out of you. Or a ton of other things I can't be arsed to list right now. As John Murdoch said, the two universes have a ridiculous amount of options going for them. Chances of Kurse being able to handle them all is very, very remote.

Is Kurse alive, he seems to be animated by Magic. That's kinda my point though, he is an extraterrestrial magical being.

I honestly don't know much about the potterverse, are there demons or such that have been affected by these kinda spells?

Kurse started off as a normal Dark Elf, and got warped into Kurse during the prison scene in TDW, by that stone-thing he was given. It's somewhat of an assumption to say he is simply animated by magic. And there are all kinds of mystical creatures in the Potterverse, ranging from dragons to vampires and more, many of whom have different magical resistances (dragons in particular, IIRC), and wizards have found ways to deal with all of them, and affected them with different magics. And in this match, those same magical creatures would also be fighting against Kurse. They also have some hax items, like time-turners, which allow people to jump back in time. There are probably a bunch of other things I am not aware of myself, as I am not as big a fan of the Potterverse as someone like Quan is.

Something else that could maybe work (though not 100% sure it would) is if a wizard grabs him, and tries to intentionally splinch (leave bits of him behind) him while apparating. It's not a direct magical attack. Plus, we have seen that his form can be damaged, so there is an argument for a wizard apparating and only taking half of him with them. Though, like I said, I have no idea if it would work. Just a theory.