Getting "The window was destroyed during method execution" error in TestExecute on windows server

That error, in my experience, is usually something having to do with an un-seen page refresh and, therefore, a timing issue... TestComplete has identified the object and so the action (click, keys, etc) is attempted but, in that split second, the object is destroyed and recreated in memory (page refresh, etc) so that the previous memory handle is no longer valid and so the click, keys, etc., generates the error.

You will need to examine the code around that particular situation and see if you can make it "smarter" to determine if the page has been properly loaded, all client-side scripts have completed, etc., before proceeding on to the UI action.

The reason why it works on yours and not on the other machine, as mentioned, is most likely timing.  The automation is running at "full speed" but on that server box there may be other factors that are slowing down the application execution so that it can't "keep up" with the automation speed.