No. Free space on a hard drive has nothing to do with power consumption. Whether the hard drive is a tradition magnetic hard drive, or a modern solid state disk, the data on the hard drive requires no power to be maintained once written to disk.
In reality, the act of deleting a file will consume more power, as the computer has to do work to accomplish the task. Granted, that amount of power is extremely minuscule. It is no different than writing on paper with a pencil. Once written, it is there permanently, however erasing it requires work.
There are a lot of ways to reduce computer power usage, however that Facebook message is false.