Here's another option - read in chunks that end with your ending text.

#!/usr/bin/perl

# https://perlmonks.org/?node_id=1216375

use strict;
use warnings;

$/ = 'APP> END_PARTU_8X8';
while( <DATA> )
  {
  s/.*APP> START_PARTU_8X8\s+APP> PARTITION_SPLIT\n//s or next;
  /(.*\n)\s*APP> END_PARTU_8X8/s and print $1;
  }

__DATA__
    APP> START_PARTU_8X8
    APP> PARTITION_NONE
    START_PARTU_16X16
    APP> END_PARTU_8X8

    APP> START_PARTU_8X8
    APP> PARTITION_SPLIT
    START_PARTU_4X4
    ........
    START_PARTU_4X4
    ........
    START_PARTU_4X4
    ........
    START_PARTU_4X4
    ........
    APP> END_PARTU_8X8
    PARTU_16X4
    16X16
    4X16
    APP> END_PARTU_8X8
[download]

Outputs:

    START_PARTU_4X4
    ........
    START_PARTU_4X4
    ........
    START_PARTU_4X4
    ........
    START_PARTU_4X4
    ........
[download]

There is a conflict between your text description of what you want printed and your output. I went by the text.