Animate rope can be used on a magical rope like an Iron Rope or a Rope of Knots - that's pretty clear. It just has to be a "nonliving rope-like object." A Rope of Knots is not living. But if we look down at the end of the spell, it says something disturbing.
The rope itself and any knots tied in it are not magical.
This is clearly to explain that if you tie a knot with the rope, it's not a magic unbreakable knot or something. But will it remove the magic of my Rope of Climbing? Will that be temporary or permanent? Does this invalidate casting the spell on my enemy's Rope of Climbing? I think not, since it states not that the spell can't be cast on a magical rope, but that the rope is not magical.
For that matter, will the rope register as magical to Detect Magic if it was not magical before? Will it register as magical to Detect Magic if it was magical before?
Yes, I am aware that this was almost certainly unintended even if it technically works - Animate Rope is not Disenchant Rope. I don't want an answer that says only "Yeah, but no sane DM would accept that." I want to know what technically happens, and then never use that knowledge.