First of all, completely uninstall Tomb Raider III, then reinstall it so that you can start again with a fresh install.
Once you've done that, download
tomb3 and drag and drop all the files into your Tomb Raider III directory, where the tomb3.exe file is. tomb3 is a decompilation project very similar to "tomb1main". It fixes many legacy bugs, restores features from the PSX version and introduces new features too.
Now download
dgvoodoo and drag and drop the "dgVoodooCpl.exe" file into your Tomb Raider III directory, the same place where you put the tomb3.exe file earlier. Also make sure to navigate into the "MS" folder, then the "x86" folder and drag and drop all of the files into the Tomb Raider 3 directory too. Now run "dgVoodooCpl.exe" and make sure to click the small button at the top that looks like this: ".\". This will make sure all the settings are saved to your Tomb Raider III directory. Now you configure the settings as you see fit, then make sure to click "apply" and "ok" when your done.
Now you can run the app called "tomb3_ConfigTool" and also configure the settings as you see fit, then make sure to click "save" to save your settings.
Now click on "Launch Setup" and for the last time, configure the settings as you see fit and press save when you're done. I recommend using a 32 bit resolution, FYI.
After that, you can click "Launch tomb3" and it should work. Enjoy.