Does the production server have ability to read, and more imprortant write files to that other server on the same network? If the 2 servers in question are not part of a domain (ad) network, then you need to create a user, say "web1" user on the web server. Then create a user on the 2nd server - same name + password, and make sure that user can then read/write files on the 2nd server.
Now, for the app-pool, use this option:
So, now when your server with IIS tries to read/write files, it will use the "user" you created on that IIS server. As long as the SAME user + password exists on the 2nd server, then you should be ok.
As noted, regardless of above, your server with IIS will need the ability to read/write/see/use files on the 2nd server. So, you could I suppose open up the 2nd server as 100% public, or add rights from ii_user to the 2nd server. However, I find in most cases, it simply better to create a logon on both servers, and give that logon the rights to the folder in question.