Inverse Fourier transform of complex exponential with frequency dependent shift
In the general case: no.
For the (inverse) Fourier transform of a function to be composed of a sum of countable diracs (i.e. to be discrete), the function needs to be periodic.
Your $e^{-j\omega\tau(f)}$ is not periodic (it can be periodic, if $\omega\tau(f)$ happen to be periodic with period rationally related to $2\pi$, but that's a pretty special case). Therefore, its inverse Fourier transform is continuous, and can't be represented by a sum of diracs.