Yes, you will need permits for this project.
Some things to look out for:
- Relocating your dryer will require moving the vent for proper air quality.
- It will likely also cause you to upgrade the electrical outlet in the new location, and possibly at the breaker box.
- Relocating your washer will require you to run sufficient plumbing to the new area and cap off the existing pipes.
- You will also need to be sure the flooring is secure in the new area, as these are heavy pieces of equipment, especially when full of water.
- You will also need a proper air vent in the room to remove humidity that can cause mold to grow in the walls or ceiling.