Diablo requires "Run as Administrator" to connect to battle.net
Diablo tries to open an incoming port when connecting to battle.net, wich is important for hosting multiplayer games. (Source: portforward.com)
Normally, a program doesn't need to have admin rights to do this, but on Vista and newer it has to "ask nicely" via UAC beforehand or an error occurs.
The problem is that Diablo (and Diablo II) were released well before Vista, so the program doesn't know what to do with the error and instead fails to connect.
This behaviour is normal if you run older games on Vista or 7, and I haven't heard of any remote exploits for that game. Even if there is one, I suspect the probability to be hit by something like this to be extremely low.
I can assure you, if the game is legitimate, it is not a security risk. If the game needs to access certain registry keys or certain files, it will need Administrator privileges.