Is it possible to run video over type-C smartphones? 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?
If your PC happens to have a WiFi adapter, you don't need to care about the connection type and can stream video content to your device via an application like Plex or Moon VR Player for example.  

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.