Ok, let's go to a step-by-step try to solve it.

Click Start, right click on "Computer" and click "Manage". You should get a UAC prompt asking you if you are sure you want to continue (Example of UAC Prompt). Click yes, and you'll be in the Computer Manager.
On the left side, click "Device Manager", and then select the + next to "Display". It should open up similar to this image (I am using a theme so colors are off, maybe a few more things). It should be displaying 2 items, either 2 NVIDIA cards or 2 ATI cards.
So, currently there are 4 scenarios.
1) You have a single NVIDIA
2) You have a single ATI
3) You have 2 NVIDIA (also known as SLI)
4) You have 2 ATI (also known as Crossfire).
Cases 1 and 2 will not be so easy to troubleshoot, especially since I have absolutely no idea what may be causing the errors :(
Case 3: 2 NVIDIA cards:
Right click on the desktop, and select "NVIDIA Control Panel", then follow the instructions here. This will allow you to disabl SLI, which should fix the error in IWD, though newer games may run a bit slower. It's up to you whether you keep it disabled permanantly, or disable it only when you want to play IWD.
Case 4: 2 ATI cards
This is a bit trickier, because it does depend a bit on the drivers you are using (aka ATI likes to move the settings around). Again, by right clicking on the desktop, you should have an option similar to "Ati Control Center", "Catalyst Control Center" or similar. It is also possible that you have that icon in the system tray, near the clock on the bottom right. In the graphics tab, it should have a setting "Crossfire", which you should then disable. Check this image for an example.

I tried to be as thorough as possible, but it is possible that I am missing a few things. Also, if you only have a single graphic card (scenario 1 and 2 above), you will probably need to talk with one of the blue people here.
Hope this helps, and apologies for the lack of a better tutorial, I haven't had any dual-gpu system (yet?).

Still no luck?

No :'(

I'm not sure if this is a video problem or not, the game doesn't appear to be freezing or having any trouble loading, it's just an error message that doesn't allow me to play.

There is only one button ('Done') on it and it returns me to the main menu, meaning even if it was a video issue, fixing that wouldn't fix the problem of playing the game!

I've combed through quite a bit of technical support posts from all over the web and never se anything exactly like this, for Icewind Dale.

The reason most of us believe this is probably a video issue is the fact that the screen doesn't get re-drawn properly. I'm not sure what the message box says, and until we manage to get it to display correctly, we can't fix it.
This image from JA2 shows a different video problem caused by using two video cards. The game does load, and you could (in theory) play it, but the graphics aren't drawn correctly. More or less same thing that happens with your game.
So, did post 61 help, did the instructions not make sense, or you didn't have time to try it yet?

P.S. Will install IWD in a while, and try to see what the message may be. We will get this game working for you, no mistakes about it.

I have a single ATI.

Not sure if this is helpful or not, but I can successfully play Planescape Torment, and Baldur's Gate 1 & 2 on this computer. (Incredibly similiar games)

I was just going to ask about this after reading this thread.

True, Baldur's Gate, Icewind Dale and Planescape all run on the infinity engine, so they should all be behaving similar. Let me see if I can find anything more on IWD.

P.S. Try running the "Config" from the start menu (GOG.com\Icewind Dale Complete\Config) and on the "Graphics" tab select "Windowed Mode", and see if running the game windowed also gives the same problem.

maybe this thread will help

http://www.gog.com/en/forum/icewind_dale/not_running_properly_on_win_7/page1

Still no luck ? :(

The problem I'm mostly having now is the inherent nature of forums. This page is getting a bit long and often times people don't spare the time to read everything so they just post a quick thoughtless response.

In turn, the page gets inflated with useless information that has already been discussed and slowly progresses through the attention spans of people clicking on the thread, making it more and more less likely that someone dedicates the time to read through everything. (I don't blame them, the amount of repeated questions & information gets tedious to sift through)

Currently I'm waiting on a response to a ticket I wrote that one of the blue's requested, and am thinking about giving up fairly soon. The Baldur's Gate Enhanced Edition is coming out soon and that would take priority over this game even if it was working.

Have you tried whitelisting the .exe in the data execution prevention list? As I recall, even GOG's version of Icewind Dale will crash on certain screens if you don't do that. It could be worth a try. Here's a thread I found with instructions on the To the Moon forum: http://www.gog.com/en/forum/to_the_moon/rgss_player_has_stopped_working

No but I promise to highlight this thread as "problem solved" and will mention it in both of my threads.

@Whitecroc : Thanks for offering but that's not relevant to the problem I'm experiencing. The game isn't crashing it's brining up an error message that I cannot read for some reason. The error has an error.

Still worth a try, I says. I've found that, on the very rare occasion DEP is the culprit, very strange things happen.