I think you're definitely on the track, to want to generalize the code to any number of directories. The first thing to do is try to eliminate variables — or rather, variable names — which perpetuate the "one chunk of code per directory" mentality.

Below is how you might reduce it to a loop.

sub Prioritise_requests
{
  my( $input_directory, $node ) = @_;

  for my $dir_num ( '1', '2', '3' )
  {
    my $input_dir = $input_directory.$dir_num;

    opendir DIR, $input_dir or die "Error opening directory $input_dir
+ - $!";
    my @files =
      sort { -M $a <=> -M $b }
      map "$input_dir/$_",
      grep /^Flat_file.*\.txt$/,
      readdir DIR;
    closedir DIR;

    foreach ( @files )
    {
      if ( Is_the_file_mine($node, $_) and Do_I_run_the_request_now($_
+) )
      {
        move $_, $input_directory or warn "Error moving $_ to $input_d
+irectory - $!";
        return;
      }
    }
  }
}
[download]

I've also made one critical bug fix in the code above: You have to add the directory name to the file name before you get any of the file's information, e.g. -M.

Also, I made a change as to the assumed return values of the two hypothetical subroutines Is_the_file_mine and Do_I_run_the_request_now. You were assuming they'd return characters. But really, they return boolean (true/false) conditions. May as well just code for that directly.