Aha! Yes, you can play 'forever'!
Gramphos's Civ3 Multi Tool program enables it.
I think you have to download the full install from the first page of the thread I am going to post, but make sure you use the latest files that he posted on this page:
http://forums.civfanatics.com/showthread.php?s=&postid=438149#post438149
His latest version (c3mto-95-1001) worked for me after I just put those files into the same folder from his previous version, allowing it to over-write any files with the same name.
Changing the ending date doesn't seem to have any effect on the number of turns you have left, however, you can change what turn you are playing. for example, I loaded a save from 1790 A.D., changed the current date to -3950 (for 3950 B.C.), and the game now considers that only 1 turn has passed, so now I have 539 more turns to play!
Edit: Yes, you can change the ending date and play more turns. It just keeps going at the 1 year/turn. So you could play 1000 turns instead of 540, by setting the retirement date at 2510 A.D. You don't notice the changes until you click on something else.
I loaded the game back up, and sure enough, it was as good as I had hoped for. Most of the world (180X180 world map) was settled, yet it was only 3950 B.C.
Just make sure you put in a correct civ3 date. I'm not sure if this causes some problems with things like when a culture improvement would get it's doubling bonus, but if you need more time to conquer the whole world, this is the way to go! (see pic below).