The flip flop operator (..) is good for this sort of thing.

use strict;
use warnings;
while (<DATA>) {
 print unless ( /^Header1$/ .. /^END LIST$/ );
}

__DATA__
Header1
1 2 3 4 5 3
2
43 43
END LIST
Header2
2 42 24 2 32 2 32
2 2  4 2 3 
END LIST
Header1
2
43 43
END LIST
Header3
2 3 4 32 
3 4 3 43 
END LIST
Header1
1 2 3 4 5 3
2
43 43
END LIST
[download]

Which can be reduced to a one liner for files:

perl -ni.bak -e 'print unless ( /^Header1$/ .. /^END LIST$/)'  filenam
+e
[download]

update: As to the way you have it.

my $text = do { local $/; <FILE> };
if ($text =~ /^Header1.*?END LIST/m) {
....
}
[download]

You are missing the /s modifier as the . will not match newlines without it. Also if you just want to remove that stuff you might as well use s/// and then print the result (add the /g modifier if Header1 appears more than once

my $text = do { local $/; <FILE> };
$text =~ s/^Header1.*?END LIST\n?//msg
[download]

Note the \n is there to remove the blank lines that might result.

-enlil