Hi there.
I know I'm a bit late to the party, but wanted to post an expanded solution (based on an answer given by pcobb23 in another forum) in case anyone else had this (very frustrating) problem. I was having the exact same issues as the above posts on my Win8.1 64bit machine, using an Origin digital download of the game (as part of a C&C multi-pack). I was able to successfully resolve the issue the following way:
SOLUTION:
Step 1. Find your Tiberian Sun install directory in Program Files ...which was in my case "Origin Games", (but others may have to look in "Westwood Studios", "EA Games" or similar if you've stumbled upon this support thread from a search engine).
Step 2. Right click on "Game.exe" application, and select properties.
Step 3. In the Compatibility tab, check the box "Run this program in compatabilty mode for:"
Step 4. Select a compatibility mode*. In my case, Windows XP (Service Pack 3) is working.
Step 5. Check the box which says "Reduce Color Mode", then set the color mode to 16-bit color.
Step 6. Also check the box that says "Run this program as an admistrator". Click apply and then ok.
Then...
Step 7. Download the below .ZIP patch file (posted by some legend from C&C Communications Centre fan forum) and simply extract the files directly into the Tiberian Sun install folder (from Step 1).
http://tore.cnc-comm.com/TS_Win8fix.zip
(I have checked the patch, it contains no malware.)
*I played around with every compatibility setting until I found one that worked... in my case it was Win XP (Service Pack 3), but I have read other forums suggesting WinXP SP2 and Win7.... if not getting the results you want, keep trying new compatibility settings! Note, as mentioned by an EA mod in another thread, this isn't an official EA patch, so he recommended backing up before trying it out.
Really hope this helps someone out there!