I have installed it and pressed Play button.
It showed some syncing process(stick to 0%) and no response.
There's no error messages.
Please let me know how can I fix this.
Thanks.

OS: Windows 10

Solved:
go to installed folder -> game shortcut icon ->
mouse right click -> property -> compatibility tab -> select window 8 or below in compatible mode

I've done this and worked, but not on GOG Galaxy. The game only open outside Galaxy.

There's a way to use galaxy to it?

Hi I have the same problem. But on Windows 11. I installed the game in my custom folder, hit Launch in GOG and nothing happened. So I tried to launch the game from the game icon in my folder but with the same effect. Then I right-clicked to game icon -> property -> compatibility tab -> select window 8 or below in compatible mode (I tried all of them form Windows95 to ti Windows10).

Nothing happenes. But I managed to find an alert window that keeps poping when I try to launch the game (both via GOG or game icon). It says:
"Unexpected program termination
Files from Heroes III are missing. Please reinstall Heroes III".

I have reinstalled multiple times but with the same effect. I am in a dead end.

Does anyone have a clue?

EDIT: I tried to run it as an administrator and I have updated DirectX just to be safe.
Procesor: Intel (R) Core (TM) i7-9750H CPU @ 2,60Ghz

You need to do what tentod said: Must be like the attached image - remember to check the "administrator" box

Then you'll need to restart both GOG Galaxy and Windows. Works for me and now I can open HoMM3 via GOG Galaxy.

That didnt help. I have the same Problem and doed this several times and that dont help

Obvious system (or 3rd party) security issues. This "security" services block Galaxy normal stability working.

To run game in admin with using gaming launchers - start gaming launcher as admin and start game directly from launcher Library.

Easier and more forrect solution steps are:
- add Games folders to exceptions of security appps if you use 3rd party ones
- reinstall everything games related to X/Games/Game-name-2/ like folders
including gaming clients(!!!) to path like D/Games/Goggalaxy/. Simple path, no System C/ drive.

- set gaming launchers exes and games (for drmfree/non-launcher games) executables to run in admin mode in exe files properties. Admin mode still need time to time and if you dont want bother yourself when not/needed - set it permanently.

Windows UAC service should be in default state: UAC promt appear after launch. Also VirtualStore should not be disabled.

In launchers always make validation after each downloading. Wo validate (verify) game installation in reinstalled and admin-mode relaunched GOG Galaxy.
Or reinstall games from offline GOG version (i doubt needed to reinstall if initially installed as GOG version, but in any way needed for GOG Galaxy - just in case). Reinstalling in Galaxy/etc are not after-downloading verification. Reinstall optional, verify mandatory.