Yes, but it won't necessarily register so unless we're talking about insanely bright distant object that should be visible through fog from any distance and which also is not a part of the skybox, this particular problem practically never arises. The flat far plane, on the other hand, is glaringly obvious in e.g. Minecraft, or any Unity game with first-person view and fog.
Now, I've googled a bit on my own, trying all kinds of search phraes, and apparently it is a known problem that the perspective projection, when wide (about 75 degrees and up) FOV is used, will distort objects at the side of the screen. One of the solutions appears to be a post-processing pass called "Panini Projection" which undoes that damage at the sides of the screen. From what I understand, it uses cylinder (but not a sphere) as the projection surface instead of a plane.
The issue you are describing now is called perspective distortion (https://en.wikipedia.org/wiki/Perspective_distortion), and it is something that also happens with physical cameras when using a wide-angle lens. There is no single correct answer for dealing with this: similarly to the situation with map projections, every projection is a compromise between different types of distortion.
Anyway, if you're writing a ray tracer it's possible to use whatever projection you want, but if you're using the rasterizer in the GPU you're stuck with rectilinear projection and any alternate projection has to be approximated some other way (such as via post-processing, like you mention).