This looks strange to me:

$file =~ m/^.*(\\|\/)(.*)/;
my $file_name = $2;

open (FILE, "$name") or die "unable to open file";
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Where is $name delcared? Oh, and the quotes around "$name" are unnecessary.

Also, what happens if somebody submits a filename like '>blah.txt'? Oops, you just deleted the contents of the file. As of perl 5.6, you can use a three-argument form of open() to avoid this problem:

open(FILE, '<', $name) or die "Can't open file: $!\n";

Better still is to do some further cleansing on the name of the file. I see you at least have some basic matching, but even better is to restrict exactly what files are allowed to be submitted via a hash:

my %allowed_files = (
    file1  => '/path/to/file1', 
    file2  => '/path/to/file2', 
    file3  => '/path/to/file3', 
);

# $file already processed from CGI params
my $file_name = defined($allowed_files{$file}) ? 
    $allowed_files{$file} : die "Bad file name\n";
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Admittedly, the above might become a maintence nightmare, depending on the number of files on your allowed list and if that list changes a lot.