while( <DATA> ) {
    print if /^Header2:/ .. eof;
}

__DATA__
<lsmothers@example.com>
SMTP
0<001501c4db9b$db8b2680$2d01a8c0@ryand9v889t9uc>
.X-Intermail-Unknown-MIME-Type=unparsedmessage
Header2: <headertwo@example.com
Received: from server.cluster1.example.com ([10.20.201.160])
        line 12

##</code><code>##

my $input = do { local $/ = undef; <DATA> };

if ( $input =~ /^(Header2:.+)/ms ) {
    print $1;
}

##</code><code>##

my $input;
{
    local $/ = undef;
    $input = <DATA>;
}
print join '', ( split /^(Header2:)/m, $input, 3 )[ 1, 2 ];

##</code><code>##

print join '', ( split /^(?=Header2:)/m, $input, 2)[1];

##</code><code>##

perl -ne 'print if /^Header2:/ .. eof' testdata.txt

perl -0777 -ne '/^(Header2:.+)/ms and print $1' testdata.txt

perl -0777 -pe '$_=join q//,(split /^(Header2:)/m,$_,3)[1,2]' testdata.txt

perl -0777 -pe '$_=join q//,(split /^(?=Header2:)/m,$_,2)[1]' testdata.txt