Combine beets, dill, olives, 1/4 cup oil, and 1 tsp. salt in a large bowl.
Cook orzo in a large pot of boiling salted water according to package directions, adding eggs 6 minutes before orzo is done. Transfer eggs to a large bowl of ice water to cool, then peel and halve. Drain orzo, reserving 1 cup pasta cooking liquid, and add to beet mixture. Stir vigorously and add pasta cooking liquid and oil as needed to fully coat. Season with salt and pepper.
Transfer pasta to a serving bowl or platter and top with eggs and feta. Serve immediately or at room temperature.
