Sure.

#!/usr/bin/perl -w

use strict;
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Always use -w and use strict. my $textfile = "iwn.txt"; Use my to declare variables under use strict.

open TEXT, $textfile or die "Can't Open $textfile";

my ($email, $id);
while (<TEXT>){  
    $email ||= $_ if /Email address:/;
    $id ||= $_ if /Message-Id:/;
}
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You were using C style looping. Perl loops are much nicer.

$x ||= $y (or equals) is the same as $x = $x || $y. The end effect for you is $id gets a value only if it doesn't already have one.

close TEXT;

print $email;
print $id;
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Pulling these variables out of the loop ensures they only get printed once.