You'll have a few hurdles to overcome as a limited user, here are the potential ones and their workaround if applicable:
Problem: Cannot install the program
Work-around: Install on external media, plug into system, play from there
Issues: Degraded performance due to the speed of the connection between device and system-bus, ability of graphics hardware, ability of host RAM and CPU.
Cannot run executables outside those specifically allowed.
Work-around: None without system intrusion.
Issues: If the computer has policy settings that only allow certain executable files to be run, then even if D3 is installed on external media, the OS will not allow it to be run. Many public computing environments block executables from removable media to cut down on viruses being carried in and installed.
Ports needed by D3 are blocked or firewalled.
Work-around: Remote Desktop to PC with privileges or Proxy server.
Issues: Those two solutions will only work if they are allowed by IT administration. Quite often, this is not the case in high security environments, as Remote Desktop is considered a security risk. Also, performance through Remote Desktop will be terrible without a very fast connection. In a work environment, use of a proxy server can often get you fired.