This modified version may be closer to what you want:

use strict;
use warnings;

my $logFileName = 'test.txt';
my $highWater = -1; # Signal first time through

while (1)
  {
  next if ! -e $logFileName;
  if ($highWater >= -s $logFileName)
    {
    $highWater = -s $logFileName; # Reset high water mark
    next;
    }
 
  open inFile, '<', $logFileName;
  seek inFile, $highWater, 0;
  #my @lines; # Uncomment to collate lines
  while (<inFile>)
    {
    print $_; # Replace with alternate code to handle output.
    #push @lines, $_; #Uncomment to collate lines
    }
  
  $highWater = tell inFile;
  close inFile;
  
  # Use @lines as required here to deal with added lines in one hit. e
+g.:
  #print @lines; #Uncomment to print collated lines
  }
  continue
  {
  sleep (5);
  }
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