What happens when you fill in all the grass area?

When you fill the grass you must place new items on storage ship. These can be moved to other islands. You will get a new island after to clear your current island at this time there are 4 islands and a 5th to be introduced in early 2021. As stated you should leave a space for bushes to be gifted. I have a 6x6 space near my mailbox where I Receive bushes and place new objects until they mature. Then I move them to their new home or to storage. I also use this space to temporarily pull things out of storage when I need more flowers to progress.

If you fill all of Orchid Island with items, you will no longer be able to receive gift bushes from visitors. There is no benefit to filling every space.