The only way I could find is to call SetErrorMode() function of Win32 API:

SetErrorMode(SetErrorMode(SEM_NOGPFAULTERRORBOX) | SEM_NOGPFAULTERRORBOX);

Unfortunately I was not able to find the way of disabling that dialog for process which we are starting (Process.Start()).

Set HKEY_LOCAL_MACHINE\SOFTWARE\Microsoft\Windows NT\CurrentVersion\AeDebug\Auto to 0

For debugging it is very useful to use followed code in Global.asax:

void Application_Error(object sender, EventArgs e)
{
    System.Diagnostics.Debugger.Break();
}

But if you deploy it on IIS then in case of unhandled exception the followed dialog window will be shown for current logged on user:

The solution is to remove System.Diagnostics.Debugger.Break(); from code before deploying.