The Portal series built a full game concept out of non euclidean spaces. Besides being great games, I think it is fascinating how true the tagline “Now you’re thinking with portals” is.
Here are two interesting experiments putting the person in different spaces than we are used to due to real world conditions. This video by Varun Ramesh demonstrates a non-euclidean ray tracer:
This other video by the MIT Game Lab demonstrates OpenRelativity, a Unity toolkit allowing simulation of navigation at relativistic speeds, used for the prototype game A Slower Speed of Light:
Update: Sylvain mentioned in comments that Carl Sagan explains those effects in the following video: