This sounds like Windows Script has been corrupted. This used to be common, and the fix was easy: just download Windows Script and install it (which basically re-installed all its components, replacing any corrupted component). But Microsoft no longer has a stand-alone installer for this. To be fair, it doesn't get corrupted very often anymore.
I don't know if this will fix it, but if I ran into this these are the steps I would follow. These steps are for Windows 7, but the process will be very similar on Vista or 8 (although I don't use 8 very often so I'm not real familiar with it yet, so could be some more significant differences there).
1) Open an "elevated" command prompt. To do this, open the Windows Start Menu and type "command prompt". Command Prompt (or "cmd.exe") should show up in the menu. Right-click on it and select "Run as administrator".
2) In the command prompt window, type each of the following lines and press "Enter" after each one:
regsvr32 msscript.ocx
regsvr32 dispex.dll
regsvr32 vbscript.dll
regsvr32 jscript.dll
regsvr32 wshom.ocx
regsvr32 scrrun.dll
After pressing "Enter" each time, a dialog should appear that says "DllRegisterServer in DLLNAME succeeded.", where DLLNAME is the name of the file registered.
After entering all of the above lines, close the command prompt window and try to open the Tournament Director again.