use strict;
use warnings;

#https://www.perlmonks.org/?node_id=525392

my $lineNUM;
while (<DATA>)
{
   
   if ($lineNUM = /^BEGIN_TAG/.../^END_TAG/  )
   {
       print "#$_" if $lineNUM =~ /^\d+$/;
       print "#\n" if $lineNUM =~ /E0/;
   }
}

=Prints
#BEGIN_TAG_110
#In this text I'm interested in.
#It can consist of several lines. Until the end tag.
#
#BEGIN_TAG_237
#I need this text.
#
=cut
 
__DATA__
Any text is in this document.
But I'm only interested in the text between a beginning and an end tag.
BEGIN_TAG_110
In this text I'm interested in.
It can consist of several lines. Until the end tag.
END_TAG and any other things I'm not interested in until the next beginning tag starts.
Any other text.
BEGIN_TAG_237
I need this text.
END_TAG
More text ...