Warcraft III is likely asking for administrative privileges so that it can download maps from other players on Battle.net. You can get around this by giving control of the program folder to the user(s) who will be running the game.

- Navigate to C:\Program Files (or wherever you installed the game).
- Right click on the Warcraft III folder and choose properties.
- Click on the Security tab.
- Click on the Edit... button halfway down the window.
- Find your user name in the pane labeled Group or user names: and click it.
- In the lower pane (Permissions for Users), click every box in the Allow column.
- Click OK twice to close the properties windows.
You may now run Warcraft III free of annoying UAC prompts.
Note that even after you have done this, you may still get prompts for admin permission when a patch has been released. I'm not sure how much you can do about that, because it's not the game asking for permission. Each patch comes in the form of a new program that Windows hasn't seen before.