TLDR: the Fourier transform is entangling.
We can immediately agree on two things:
- if you input a computational basis state (separable) to the Fourier transform, it outputs a separable state
- the circuit involves entangling gates
Neither of these actually resolves the question. Could there be another separable basis which is converted to entangled states? Could entangling gates combine to give an overall separable operation (example: swap gate being composed of 3 controlled-nots)?
The best way to resolve this is with an example. So, take the simplest example: the Fourier transform on two qubits. The circuit looks like this:
This is easy to analyse because there's only one entangling gate: the controlled-$S$. Any single-qubit gates that come after it do not affect entanglement. Now, if controlled-$S$ acted on an input of $|+\rangle|+\rangle$, it would create entanglement. So, we can work backwards: if I input $(H\otimes I)|+\rangle|+\rangle=|0+\rangle$ to the circuit, the output must be entangled. Separable input yields entangled output, so the circuit is entangling.