Extents problem. What I mean by that is that the image is being displayed but the Full Extent command is taking you to a zoom level which is too far out (in your case you have 1 : 1,238,063). Can you zoom to the area where you know that the image should be displayed at a reasonable resolution?
Projection (spatial reference) problem. Is the coordinate system associated with the image correct?
Image service creation problem. When you created the image service did you take the steps needed to remove all the warnings, e.g. generate statistics, overview, pyramids, etc.? You did not describe the steps that you followed to create the image service.
When ArcMap connects to ArcGIS Server you connect as a user. Is the production machine connecting using the same user as the machine that works?
I am guessing that this is some type of permission issue.
Here in Esri support, we'd ask you to install Fiddler on these machines, start it up, and reproduce this issue. You could then compare the successful result against the unsuccessful result and see where the failure is.