Batch File - easy!
I've funnily enough had this problem with Galactic Battlegrounds too.
A simple .bat file which can be created in a moment can fix having to delve in and out of the Task Manager.
Open up Notepad and type in the following (with line feeds):
taskkill /F /IM Explorer.exe
Save that to the Desktop but instead of Saving As Text Document, click the dropdown and select All Files. Then call the file TIE.bat
Cut and paste from your Desktop to the TIE Fighter installation folder (by default C:\Program Files\LucasArts\TIE95. You may get told you need Administration Privileges, just go right on ahead and hit Continue.
You can modify this batch file for ANY game that throws up bizarre colour issues. Simply change the second line each time to the relevant .exe for the game you're trying to play (e.g. in the case of Battlegrounds it read Battlegrounds_x1.exe instead of TIE95.exe.
You can even, if you're feeling very clever, replace any shortcuts you have in your Start Menu or on the Desktop to the .BAT file itself, meaning you don't have to go hunting for it each time you want to play.
The good news is when you click it, it'll stop Explorer fully in its tracks and run the game as uninterrupted as possible. The .BAT then waits for TIE95.exe to finish (i.e. for you to close it when you're done playing.) The next line will then cause explorer.exe to re-run for you so you come back to your full desktop.
Hope that helps!