What is your TB version, and which add-ons are installed? Some, such as ReminderFox, interfere with tag colors. See if the tags hold their settings in safe mode (hold Shift when TB is launched).

I have version 78.1 but the issue is there in 68.11, too. The only extension installed is the Color Folders extension, and this issue happens whether that extension is install or not.

Are the problem tags the 5 built-in tags or custom tags? Is the profile you're running in 78 the same one that was used in 68? I don't have any issues with tags in 78, but I created a new profile with 78 and only transferred some parts of the old 68 profile manually. You could create a new profile in Help/Troubleshooting, about:profiles, add an account, and see if tags work properly. Also, try running in safe mode as suggested before; it involves more than just disabling add-ons.

Yes, the 5 built in tags. We have 1 of 4 design computers (running 68.11) that holds the tag colors. The other 3 won't hold (1 on 68.11 and the other 2 on 78.1).