http://forums.ubi.com/showthread.php/227905-Heroes-of-Might-and-Magic-III-sigh-Forums
It seems the issue isn't related to the tenth version; the post dates back to 2007... And yes, the solution seems to have been found in the latest posts. It's a similar problem.
If you think the problem is related to file writing permissions, you should simply run Heroes III with administrator privileges.
Here's another link: http://www.justanswer.com/computer/6m2q0-i-m-loading-old-game-software-heroes-iii-following.html#re.v/328/
Okay, in reality, there are different problems everywhere if you look closely, but they are similar: everywhere the problem is in the C++ libraries.