ITAGUAÇU, BRAZIL—Even as the rains begin to fall, the drought that forced Brazilian farmers like Carlos Babilon to dig up their coffee trees is set to continue to affect prices in the coffee market.

Mr. Babilon has uprooted about 8,000 coffee trees on his farm in Espirito Santo, a crucial Brazilian coffee producing state that harvests the robusta bean that goes into much of the world’s instant coffee. Prolonged drought here has meant the tress were too parched to save, especially the older ones, Mr. Babilon, a fourth-generation...