Mine runs without that problem and I'm on Windows 7 64-bit.
(Edited to include OS details.)

I also had no such problems.

You mention the steam version, if you were running the steam version and now you are running the GOG version, might want to uninstall the steam version. With both installed you might run into some problems.
Check the path for the game shortcut just to be sure you're running the GOG install.

I installed the GOG Tropico Reloaded (I don't have steam) and I am getting the same error message shown in the OP when I try to run Tropico 1
On Win XP Pro

Nah, it's long gone and I've reformatted since then by installing 7. Don't exactly know what I could do, searching for this error comes up with people looking for no cd hacks on torrent sites.

Best I can do is direct you to the GOG support, you probably already done that so . . .
http://www.gog.com/en/support/contact

Yeah, I'm waiting on that. Strange how this game was so anticipated at GOG yet this forum is almost empty. I wonder how many other people are having my problem...

I just finished downloading the game yesterday. I had the same problem when I tried to run the game. I'm on a Win XP Home Edition. 768 MB RAM and a 128MB video card. The computer's almost as old as my marriage (6 years) so I first thought it was just my computer acting up. Looks like it might not be.
What I did was to press on the continue button twice and then the game would load. It's not game breaking though it would be nice not to have to see the error popping up every time I play the game.

I was thinking about getting TR, came here to read comments . . . not much to go on actually.

impulse had a tropico sale which might have something to do with it, i had no idea it was coming to GOG and bought t. reloaded then

It's been a few years, but I seem to recall having 'launch' problems back when I first owned Tropico 2 - along the lines of having to double click continue. Currently downloading, and will install/launch in awhile (I'm running XP on an ancient Lenovo).

I had the insert CD message with tropico 1 as well, but this was with wine, so I can't really complain.
Tropico 2 works fine though

I don't have a continue button in the error message, just an OK. And double clicking that will do nothing.

Actually installed the game late last night, and both T1 and T2 run fine. I remember now the problem I had with the original version I owned: The cursor would jump around wildly (possibly due to my ancient W-98se machine with a low-end video card). But so far I haven't had any major problems.