It looks to be because you're not constructing the full path of the file.

After you read the list, make sure that you prepend the directory name to each file before trying to open it:

my $dir = "start";
foreach my $file (@files) {
    my $path = "$dir/$file";
    open FILE, $path or die "Can't open file '$path': $!";
    while (<FILE>) {
        print $_;
    }
}
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Update:  Now that I re-read your question, I realize I missed that it's the directory you're having trouble with (although my comments above will still apply, once you've read the directory successfully, since you are, in fact, trying to open each file, and that will require the full pathname).

If you're doing this through http, try having the script print the working directory (after, of course, printing the Content-type header):

use Cwd;
my $thisdir = getcwd;
print "Content-type: text/html\n\n";
print "Current directory = $thisdir\n";
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You should be able to figure out how to traverse into cgi-bin relative to that.  One guess is that it might be simply /cgi-bin.