What you have is on the right track, but you might want to take a few other precautions:

#!/usr/bin/perl -w

use strict;
use CGI::Carp;
use CGI;

my ($cgi) = new CGI();

my ($gif_name) = 'whatever.gif';  # Or $cgi->parameter('show')
my ($image);

open (GIF, $gif_name) || die "Could not open $gif_name: $!\n";
binmode (GIF);
read (GIF, $image, -s $gif_name);
close (GIF);

binmode (STDOUT);

print $cgi->header("image/gif");
print $image;
[download]

Note the use of '-s' to find the complete size of the file, which allows you to read it in one go.

CGI::Carp will allow you to record any run-time errors to the appropriate error_log file, especially those messages generated when you 'die', which normally go sight-unseen.

Update: Added 'binmode(GIF)' as pointed out by dws below, and merlyn as well (Thanks!) No sense in posting 'bad' code.