The issue with Chained to the Rocks here is the text on the card itself "Enchant Mountain you control." This means that whomever controls Chained to the Rocks must also control the mountain it is enchanting. If you do not meet that criteria when you check state-based actions, it falls off (sort of like giving a creature protection from white causes Pacifism to fall off, or, more pointedly, how pumping a creature's power causes Runner's Bane to fall off).
Now, can you use this for some permanent exiling shenanigans like people have done with Sundial of the Infinite and whatnot? My initial reaction would be no, as any modification of ownership of either Chained or the Mountain would shift and the spell would be invalid immediately (not going on the stack). My understanding, and I'm not a judge by any means, is that as the stack resolves, each layer progresses through player priority and the checking of state-based effects before any effects would resolve.
In other words, if you would cast Chained and exile a creature on its ETB trigger, and then use Zedruu to swap ownership of either Chained or the Mountain, the instant that ability resolves, Chained falls off and they get the creature back. Similarly, if you were to cast Chained and try to swap control of the mountain you targeted before it resolves, that's also a no-go, as it no longer has a valid target and it fizzles with nothing having ever been exiled.
So the tl;dr version - no, you cannot exile a creature permanently with Chained beyond just keeping it on your mountain and hoping they don't have any enchantment hate.