filters – Does FFT filtering preserve the phase shift of a PSK signal?

“FFT filtering” is just filtering (if not buggy); it’s the actual filter that defines the signal properties of that. In other words, it’s the filter coefficients, and not the algorithmic shortcuts you take.

In communications we typically use linear-phase filters, since they have constant group delay and hence don’t introduce any phase distortion.

So, if you use an appropriate filter design, and use a correct implementation of a convolution algorithm, then, there will be no problem with your PSK phase.

if we cancel out all other frequencies besides the desired frequency by multiplying them by zero

That, however, is not how you build a convolution algorithm. See this very popular answer to the question “Why is it a bad idea to filter by zeroing out FFT bins?”.

A note on computation: I’m all for accelerating of computations, but isn’t TNC something with single- to two-digit kilohertz bandwidths and relatively large channel spacing? I don’t really think anything computationally stronger than a toaster will benefit much from calculating the FFTs for these channels on a GPU (overhead of getting signal in and out of the GPU will outweigh the power saving compared to doing the same computations on a CPU); I’d expect that at these rates, a power-efficient method is actually to compute fast convolution (“FFT filtering”) plain on the CPU. I might be wrong, and you need to have extremely steep frequency responses for some reason, but considering the age of everything involved, I’d be more than surprised if that was actually a way to get better signal out of it.