I have a unfinished basement. I need to move a hard wired smoke detector to a different location. What will happen when I cut power to the entire house? Will all detectors start chirping?

They should if Batteries are good, When you shut power off, remove batteries to stop the chirp, reinstall batteries when powered back up, batteries may need replacement if you don't remember how old they are.
Don't worry about low batteries too much, as they will chirp for low battery as well.

Why cut power to entire house? Why not just the smoke detector circuit.