Rope and tent

Jim-

You don't need rope, as there are no technical sections on the GR20. On sections that are very steep there are chains to assist you, if you need them. I've seen people use rope and carabiners in the Cirque de la Solitude, although most people do not feel they need them.

If your plan is to stay outside, you should bring a tent because in June the weather can be cool at night and there is always the possibility of rain. If you want to stay in the refuges, then a tent is not necessary. But even in June the refuges can fill up, leaving you with fewer options than if you carry a tent.

My advice: leave the rope at home, bring the tent. And have a great time. June is beautiful in Corsica.