From the Super Mario Wiki
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When Mario and Luigi start their search for Eldream's Nightmare Chunk, they soon find out that Dreambunny has it but will not give it away. After some more failed attempts to catch the rabbit, the Bros. run into Dreamy Brickle who also wants to catch Dreambunny in order for his theme park to gain popularity. The Bros. eventually get the chunk by chasing the rabbit into a dead end, making Dreambunny willing to give it up. They shatter the Nightmare Chunk, which reveals Eldream. The Pi'illo states that Dreambunny is his pet, and after a reunion, decides to help the Bros. by opening up a portal to Dream's Deep. Dreambunny assists him in this task by drilling into the ground and drilling out the same way, thereby creating the portal. It later helped get an unconscious Mario, Luigi, and Princess Peach back to the real world after defeating Antasma and Bowser in Dream's Deep. Dreambunny is not seen again for the rest of the game.
Pi'illoper's book suggests that sometime before the events of the game, when the Pi'illos were put into a deep sleep, Dreambunny saw Eldream trapped in the chunk, and as if trying to protect the Pi'illo, the rabbit lay against the stone, closed its eyes, and slept.