While your answer is accurate to within the posted spec, for good form it's probably better to make it new line tolerant and to encourage people inlining variables into regexes to escape meta characters.

#!/usr/bin/perl

use strict;
use warnings;
use diagnostics;

my $start = '<START>';
my $end   = '<END>';

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

while ($data =~ /\Q$start\E(.+?)\Q$end\E/sg) {
    print "$1\n";
}

__DATA__
<START>TEXT1<END>
<various data>
<START>TEXT2<END>
<various data>
<START>TEXT3<END>
[download]

If they are concerned about holding the whole file in memory, there is a convenient choice for record separator:

#!/usr/bin/perl

use strict;
use warnings;
use diagnostics;

my $start = '<START>';
my $end   = '<END>';

local $/ = $end;
while (<DATA>) {
    while (/\Q$start\E(.+?)\Q$end\E/sg) {
        print "$1\n";
    }
}

__DATA__
<START>TEXT1<END>
<various data>
<START>TEXT2<END>
<various data>
<START>TEXT3<END>
[download]

where I've kept the regex as is since the last record will not be <END> delimited, and so there'd be a failure for an unmatched <START>