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.

#!/usr/bin/env perl
use Modern::Perl;

open my $labels, '>', 'labels.txt';
open my $tables, '>', 'tables.txt';

{
  $/ = '';
  START:
  my $record = <DATA>;
  exit unless $record;
  my $fd = $record =~ /\*\s+DC/s ? $tables : $labels;
  print $fd $record;
  goto START;
}
__DATA__
LABEL#1 EQU *
     $MAC ABORT 

LABEL#2  EQU   *                                                      
+          
     $NOT UPDATE,STOP                                    

T#TAB1   EQU   *                                                      
+          
   DC    AL4(-1)
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