The problem with the local textures I had in the beginning also.
Then I tried and found out that only Firefox works. The fox obviously recognizes when the page is loaded from the same location and allows loading the graphic. Other browsers give an error message!
I use Windows ( 64, Firefox 57)
Try a simple example of me. They all run locally at my place.
http://threejs.hofk.de/BufferGeometry/02_buffer.html
http://threejs.hofk.de/
UPDATE: This no longer applies to Firefox since 2019.
Because there was now a heart for it, attention was drawn.
Other possibility see
https://hofk.de/main/discourse.threejs/
( post Collection of examples from discourse.threejs.org )
read
https://hofk.de/main/discourse.threejs/Local%20use%20of%20the%20examples.pdf