It won't be very good for Real Madrid if they become out of the Copa del Rey as well. Because they have already lost most of their chance to win the La Liga title. If they get eliminated by Barcelona in the Copa del Rey as well they will have only the Champions League left. Finishing the season without a title would be really bad for Real Madrid.
Barcelona can really push them out of the Copa del Rey by the way. Because they already beat them in the Santiago Bernabeu in the first leg. I think they would finish the incomplete job.
They've already let the La Liga title slip away from them, and they're on the verge of losing the Copa del Rey as well. If Barcelona eliminates Real Madrid from the Copa del Rey, which seems likely, Real Madrid's main focus will be on the Champions League, and they will do everything possible to finish this season with at least a league title. They already have a good reputation there, and they'll want to maintain it while adding to their number of champions league titles.