DynamicDD: I have the same problem. Not after a short while, but the game is just instantly lagging - especialyl once i enter teh screen of teh advisors. I haven't tried out if it gets even worse after while.
I'm on win 10 with integrated graphics.
I have actually civ3 on steam and there it runs fine, except for the video-playback-/victoryscreen bug, where the game will crash upon mouseclick.
That's why i bought it here on gog but even though I don't have the bug from the steamversion anymore, the lag here makes it even worse.
The "lsansfont" error that OP1 mentioned didn't work for me.
The gog civ3 also looks a bit more "pixelated" / sharp-edged than the steam version - so my amateurish guess is, that it has sth to do with the compability with my graphics-chip.
Regarding the "lsansfont" error, did you delete or rename the LSANS.FOT file before running the game? The LSANS.FOT should regenerate after the start-up but this should fix the problem. If the problem persist try to open the LSANS.FOT with notepad and edit it so that the path to the font file is correct.