Is there any practical application for performing a double Fourier transform? ...or an inverse Fourier transform on a time-domain input?
No, taking the Fourier transform twice is equivalent to time inversion (or inversion of whatever dimension you're in). You just get $x(-t)$ times a constant which depends on the type of scaling you use for the Fourier transform.

The inverse Fourier transform applied to a time domain signal just gives the spectrum with frequency inversion. Have a look at this answer for more details.