Your data doesn't match your specification...

You have data items ending with  etc but your narrative specifies "beginning with TAG1 *and* ending with TAG2" (and your code -- very slightly modifed with extra "say" and "use" statements -- at a quick glance seems to [more-or-less] support that narrative).

#!/usr/local/bin/perl
#use strict;
use warnings;
use 5.016;

# 1071296

while (<DATA>) {
# This section works OK.
    say "Positive lookaround first: ";
    if (/(?<=TAG1)(.*)(?=TAG2).*?/x) {
        print $1, "\n";
    }

   if (/^(?<!TAGS)(.*?)(?=TAG2).*$/x ) {
       say "\t Negative lookaround:";
       print $1, "\n";
   }
}

# Data now matches narrative; last item, stet, as out-of-spec test

__DATA__
TAG1 text one TAG2
TAG1 text two TAG2
TAG1 text three  TAG2
TAGS text four TAG2
TAG1 text five TAGT
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OUTPUT:

Positive lookaround first:
 text one
         Negative lookaround:
TAG1 text one
Positive lookaround first:
 text two
         Negative lookaround:
TAG1 text two
Positive lookaround first:
 text three
         Negative lookaround:
TAG1 text three
Positive lookaround first:
         Negative lookaround:
TAGS text four
Positive lookaround first:
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Updated to correct output shown. /me cut'n'pasted the wrong run, lacking tabs for ease of scanning.