Taking toolic's suggestion, I did this, stealing fairly directly from the Synopsis, and as seen below, to good effect.

Three points of note:

1. When using the text file technique, the lines appear to come back either pre-chopped or pre-chomped;
2. The internal line-counter feature is 1-based, not zero-based, and;
3. As written, it may have added an extra line to the end of the file. I have examined neither the accuracy of this supposition, nor its cause (if real). That is left as an exercise for the reader. If it is real, that might be due to the way the while()loop is constructed for reading the file.

#!/usr/bin/perl
use strict;

use Archive::Zip;
use Archive::Zip::MemberRead;

my ($zipfnm, $mbrfnm, @arglst) = @ARGV;
{
     if (-e $zipfnm)
     {
          # ZIP file exists.  Proceed.
          my $zipobj = Archive::Zip->new($zipfnm);          # ZIP obje
+ct pointer
          # Adjust Windows specifications
          my $mbranm = $mbrfnm;
          $mbranm =~ s/\\/\//g;
          # Proceed.
          my $mbrhan = Archive::Zip::MemberRead->new($zipobj, $mbranm)
+;
          # Synopsis does not specify error handling return;
          # Left as an exercise for the reader :-)
          # OP indicates reading lines from the file, so use the text 
+file technique
          while (defined(my $inpbuf = $mbrhan->getline()))
          {
               # Note the line read is already chopped or chomped (!!)
               print "Line " . $mbrhan->input_line_number . ": [$inpbu
+f]\n";
          }
     }
}

exit;

__END__
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Results:

D:\PerlMonks>"C:\App\7-Zip\7z.exe" l -tzip test.zip

7-Zip [64] 9.20  Copyright (c) 1999-2010 Igor Pavlov  2010-11-18

Listing archive: test.zip

--
Path = test.zip
Type = zip
Physical Size = 17540

   Date      Time    Attr         Size   Compressed  Name
------------------- ----- ------------ ------------  -----------------
+-------
1995-03-03 22:42:08 ....A        24361        16886  binary.dat
2015-03-10 15:40:58 ....A           22           22  test.dat
2015-03-10 15:41:09 ....A           28           28  test2.dat
2015-04-23 19:41:25 ....A           82           62  test3.txt
------------------- ----- ------------ ------------  -----------------
+-------
                                 24493        16998  4 files, 0 folder
+s

D:\PerlMonks>type test3.txt
This is line one.
This is line two.

This is line four.  The third was blank.

D:\PerlMonks>zipreadfile1.pl test.zip test3.txt
Line 1: [This is line one.]
Line 2: [This is line two.]
Line 3: []
Line 4: [This is line four.  The third was blank.]
Line 4: []

D:\PerlMonks>
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