The Moon and the Earth are already roped together—gravitationally. And we know what that means.

I gather the question is if another, physical tether were attached to both. It'd most likely have a healthy amount of slack to account for the elliptical orbit. Perhaps a counterweight is in order at some intermediate point to provide tension. In any case, the rope would have to endure its own weight only (which is still quite a task!), plus some for doing useful work.