When I choose Multiplayer then LAN I get the error message:
"No multiplayer support present. You have been switched to viewing only local games."
When I Google this i get 0 hits.
In the next screen I can host a game and others can join and we can play just fine. I then saved the game before any players quit. When I went to load the game again it loaded as a single player game and there is nowhere in the Multiplayer LAN process to choose loading a game.
Any idea what is going on? Are these two seperate issues or is the error message preventing me from loading a game in LAN mode.

I got this message, too, but you can ignore it. The GOG version supports Direct IP, and it works!

If you start a multiplayer session for the first time, you will be asked to install DirectPlay. I did this and restartet Civ3.

After that, I was able to host a session. On the client PC, I clicked on multiplayer > Direct IP and entered the IP address of the host. I played two turns on both machines, saved the game on both machines, quit, startet Civ3 again, and both machines could load their old game.