Well there are lots of variables.
Do they 'have' to give up, in this scenario, because their partner no longer wants to have an open marriage/poly set up? In that case, there are two options - either being poly is fundamental to how they approach relationships, in which case, no, I imagine the relationship wouldn't survive because they'd no longer be compatible. Or being poly isn't a dealbreaker for them and they're happy to close the relationship again, in which case there's no reason the marriage wouldn't survive.
If they 'had' to for other reasons - health, logistics, whatever - then again I can't see why the marriage wouldn't survive.
As you've said in your post, you don't understand the drive for an open relationship, so perhaps it's difficult for you to understand, but lots of people have happy, thriving poly or open relationships, which aren't necessarily about an obsession with sex (and even if they were, we're kind of biologically wired to want sex - there's nothing inherently strange about that).