Forbidding the use of a while loop seems like an odd homework requirement, since a while loop is the idiomatic way to read data in Perl. But you're welcome to use this 100% while-free solution:
UPDATE: I see that the requirements changed, and records can no longer be assumed to be divided by a blank line. But the point remains: rather than reading line-by-line and trying to succeeding lines based on previous ones, it's often much easier to read block-by-block, if it's possible to find a way to set the input record separator to do so.
open my $labels, '>', 'labels.txt';
open my $tables, '>', 'tables.txt';
$/ = '';
my $record = <DATA>;
exit unless $record;
my $fd = $record =~ /\*\s+DC/s ? $tables : $labels;
print $fd $record;
LABEL#1 EQU *
LABEL#2 EQU *
T#TAB1 EQU *
Available for small or large Perl jobs; see my home node.