Is it possible to run a video from a PC (over type-c, hdmi, or displayport) to a type-c smartphone for VR? What limitations would be faced?
The limitations are inherent to the video player you choose to use on your phone.
I assume you are interested by VR headsets since you included the tage "samsung-gear-vr".
For Plex for example, there is currently no native support for Gear VR, so you have to access the web interface by Oculus browser or Samsung Internet browser (it also seems to be supported by PhoneCast). You can also use a more complex setup such as the one described in this post.