Rules-As-Written: Nothing happens
The rules for the spell state that the ground is covered in a slippery substance, and that it confers Saving Throws against falling prone for creatures in the area. It doesn't say anything about being applied to objects or, per your example, ropes used for climbing, nor does potentially being knocked prone have an effect on creatures who are clinging to a rope.
Slick grease covers the ground in a 10-foot square centered on a point within range and turns it into difficult terrain for the duration.
When the grease appears, each creature standing in its area must succeed on a Dexterity saving throw or fall prone. A creature that enters the area or ends its turn there must also succeed on a Dexterity saving throw or fall prone.
—Grease, Player's Handbook, pg. 246
It's probably necessary for the DM come up with an explanation of what should happen in this situation
Personally, as DM, I don't think I would approve of simply saying "nothing happens". This is a clever use of a spell in a manner that the rules aren't designed to accommodate, so I think it's appropriate to allow it to have some kind of effect. In this context, I would likely allow 10' of rope to be covered in grease, and for creatures that try to climb across it, I would force them to make Dexterity Saving Throws to stay affixed to the rope.
But that is, ultimately, a DM's call, and there's no twisting of the game's rules-as-written to permit this kind of use.